[Paper Review] A piecewise contractive dynamical system and election methods

Svante Janson, Anders Öberg|arXiv (Cornell University)|Sep 19, 2017

Mathematical Dynamics and Fractals15 references5 citations

TL;DR

This paper analyzes a piecewise contractive dynamical system on the unit interval with a discontinuity, proving the existence of a universal limit cycle for non-exceptional parameters and showing that the exceptional parameter set has Hausdorff dimension one and finite measure under a specific gauge function. The results are applied to Phragmén’s and Thiele’s multi-winner election methods, demonstrating that Phragmén’s method yields rational seat proportions in the limit (except on a tiny set), while Thiele’s method can yield irrational limits, implying no typical periodicity.

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 Phragmén 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 Phragmén's method.

Motivation & Objective

• To study the dynamics of a piecewise linear, contractive map on [0,1] with a discontinuity at points where two values are taken.
• To characterize the invariant set, limit set, and rotation number of the system, especially distinguishing rational and irrational cases.
• To prove that the exceptional parameter set—where the rotation number is irrational—has Hausdorff dimension one and finite measure under the gauge function h(t) = 1/|log t|².
• To apply the dynamical system results to Phragmén’s and Thiele’s multi-winner voting methods, analyzing the convergence of seat proportions.
• To show that Phragmén’s method yields rational limit seat shares almost everywhere, while Thiele’s method can yield irrational limits, implying no typical periodicity.

Proposed method

• The system is defined by two functions: f₋(x) = {ax + b} and f₊(x) = {ax + b}₊, where the latter takes value 1 instead of 0 when ax + b is integer, introducing a discontinuity.
• The analysis uses symbolic dynamics, lifts to the real line, and defines a rotation number via orbit averages, proving its existence and convergence for all orbits.
• The paper employs gauge functions and Hausdorff measure to quantify the size of the exceptional parameter set, showing finite measure for h(t) = 1/|log t|² and positive measure for h(t) = 1/|log t|.
• It proves the existence of a unique invariant measure and classifies orbits based on periodicity and discontinuity inclusion.
• The election method application maps the dynamical system to seat allocation: each iteration corresponds to seat assignment, and convergence of orbit points corresponds to stable seat proportions.
• For Phragmén’s method, the system is shown to converge to a rational limit proportion for each party, except on a set of parameters with Hausdorff dimension one.

Experimental results

Research questions

• RQ1 Does the piecewise contractive dynamical system with a discontinuity admit a universal limit cycle in non-exceptional cases?
• RQ2 What is the Hausdorff dimension and measure of the set of parameters for which the rotation number is irrational?
• RQ3 How does the dynamical system model Phragmén’s and Thiele’s multi-winner election methods, and what are the limiting seat proportions?
• RQ4 Can the limit seat proportions in Phragmén’s method be irrational, or are they always rational?
• RQ5 Does Thiele’s method exhibit typical periodicity in seat allocation, or can the limit be irrational even in typical cases?

Key findings

• The system has a universal limit cycle in non-exceptional cases, meaning all orbits converge to the same periodic orbit.
• The exceptional parameter set—where the rotation number is irrational—has Hausdorff dimension one and finite Hausdorff measure under the gauge function h(t) = 1/|log t|².
• The set of parameters with irrational rotation number has positive Hausdorff measure under the gauge function h(t) = 1/|log t|, indicating it is not arbitrarily small.
• For Phragmén’s method, the proportion of seats for each party converges to a rational number in the limit, except for a set of parameters with Hausdorff dimension one.
• In contrast, Thiele’s method can yield irrational limit seat proportions even for typical parameters, implying no typical ultimate periodicity.
• The paper proves the existence of a unique invariant measure for the system, which is crucial for understanding long-term statistical behavior.