Alternate Scaling Algorithm for Biproportional Divisor Methods

  • In parliamentary elections biproportional divisor translate votes into seats so that, for each district, fixed seat contingents are met and that every party receives as many seats as the overall vote counts reflect. A set of district-divisors and party- divisors ensures that proportionality is respected both within the districts and within the parties. The divisors can be calculated by means of the 'alternating scaling algorithm' (AS-algorithm) which is formally introduced. It is the discrete variant of the 'iterative proportional fitting procedure' (IPF-procedure). The AS-algorithm iteratively generates scaled vote matrices that after rounding alternately fulfill the district-contingents and the party-seats. Thus it defines two sequences: the AS-seat-sequence and the AS-scaling-sequence. The central question in this paper is under which condition the AS-algorithm is able to generate the set of biproportional apportionments. The conjecture of Balinski & Pukelsheim (2006) is provenIn parliamentary elections biproportional divisor translate votes into seats so that, for each district, fixed seat contingents are met and that every party receives as many seats as the overall vote counts reflect. A set of district-divisors and party- divisors ensures that proportionality is respected both within the districts and within the parties. The divisors can be calculated by means of the 'alternating scaling algorithm' (AS-algorithm) which is formally introduced. It is the discrete variant of the 'iterative proportional fitting procedure' (IPF-procedure). The AS-algorithm iteratively generates scaled vote matrices that after rounding alternately fulfill the district-contingents and the party-seats. Thus it defines two sequences: the AS-seat-sequence and the AS-scaling-sequence. The central question in this paper is under which condition the AS-algorithm is able to generate the set of biproportional apportionments. The conjecture of Balinski & Pukelsheim (2006) is proven stating that the AS-algorithm is effective for all biproportional apportionment problems that come with at most a few ties. In the rare event that the set of biproportional apportionments cannot be determined by the AS-algorithm, the complementary AS-Tie&Transfer-combination puts things right. Its analysis leads to a constructive proof of necessary and sufficient conditions for the existence of biproportional apportionments. If these conditions are violated, the sequences generated by the AS-algorithms may have more than two accumulation points. On the contrary, the IPF-procedure has at most two accumulation points.show moreshow less

Download full text files

Export metadata

Statistics

Number of document requests

Additional Services

Share in Twitter Search Google Scholar
Metadaten
Author:Kai-Friederike OelbermannGND
URN:urn:nbn:de:bvb:384-opus4-22680
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/2268
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2013-04)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Release Date:2013/03/12
Tag:alternate scaling algorithm; biproportional representation; biproportional electoral system; divisor methods; iterative proportional fitting; controlled rounding
GND-Keyword:Statistik; Wahlverfahren; Skalierung; Iteration; Rundung; Kontingenztafelanalyse
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht mit Print on Demand