A piecewise contractive dynamical system and election methods

TL;DR

This paper analyzes a piecewise linear, contractive dynamical system on the interval, proving the existence of universal limit cycles, characterizing exceptional parameter sets, and applying these results to election methods to determine seat proportions.

Contribution

It introduces new results on the dynamics of a piecewise contractive map, including limit cycles and invariant measures, and applies these findings to analyze election methods and their convergence properties.

Findings

Existence of a universal limit cycle in non-exceptional cases

Exceptional parameter set has Hausdorff-dimension one

Proportions of elected seats converge to rational limits in Phragmén's method

Abstract

We prove some basic results for a dynamical system given by a piecewise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we give a thorough investigation of the dynamics; studying the cases of rational and irrational rotation numbers separately, and we show the existence of a unique invariant measure. We apply some of our results to a combinatorial problem involving an election method suggested by Phragm\'en and show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters. This is in contrast to a related election method suggested by Thiele, which we study at the end of this paper, for which the limit can be irrational also in typical cases and hence there is no typical ultimate periodicity as in the case of Phragm\'en's method.