## 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.