Voting with rubber bands, weights, and strings

We introduce some new voting rules based on a spatial version of the median in R known as the mediancentre, or Fermat-Weber point. Voting rules based on the mean include many that are familiar: the Borda Count, Kemeny rule, approval voting, etc. (see [35], [36]). These mean rules can be implemented by “voting machines” (interactive simulations of physical mechanisms) that use ideal rubber bands to achieve an equilibrium among the competing preferences of the voters. One consequence is that in any such rule, a voter who is further from consensus exerts a stronger tug on the election outcome, because her rubber band is more stretched. While the R median has been studied in the context of voting, mediancentrebased rules are new. Voting machines for these rules require that the tug exerted by a voter be independent of his distance from consensus; replacing rubber bands with weights suspended from strings provides exactly this effect. We discuss some novel properties exhibited by these rules, as well as a broader question suggested by our investigations — What are the critical relationships among resistance to manipulation, decisiveness, and responsiveness for a voting rule? We argue that a distorted view may arise from an exclusive focus on the first, without due attention to the other two. 1 Rubber band voting machines We’ll consider voting rules for which A is a finite set of alternatives, and a ballot consists of a strict ranking (linear ordering) of A’s members. Let’s start with A = {p, q, r}, so there are six possible ballots. Using any standard web browser (Safari, Firefox, or Explorer, for example) navigate to the Evolver simulation at http://www.math.union.edu/locate/voting-simulation Click on the link Voting with rubber bands, weights, and strings. You will find a regular hexagon, whose vertices are labeled with these six ballots, as in Figure 1, along with links to other pages. Make sure that the Model button under the hexagon is toggled to the Mean position.