Voting with rubber bands, weights, and strings

We introduce some new voting rules based on a spatial version of the median 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 Zwicker (2008a,b)). 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-1 median has been studied in the context of voting, mediancentre-based 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. (C) 2011 Elsevier B.V. All rights reserved.