Divisor-Based Biproportional Apportionment in Electoral Systems: A Real-Life Benchmark Study

Sebastian Maier, Petur Zachariassen, Martin Zachariasen
2010 Management science
Biproportional apportionment methods provide two-way proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row-and column-sum requirements are fulfilled. In a divisor-based method for biproportional apportionment the problem is solved by computing appropriate row-and column-divisors, and by rounding the
more » ... rounding the quotients. We present a comprehensive experimental evaluation of divisor-based biproportional apportionment in an electoral system context. Firstly, we study the practical performance of a range of algorithms by performing experiments on real-life benchmark instances (election data with multi-member districts). Secondly, we evaluate the general quality of divisor-based apportionments with respect to, e.g., deviation from quota, reversal orderings and occurrences of ties. 1 to parties r = (r 1 , r 2 , . . . , r n ) is computed by using any of the classical vector apportionment methods (see [9] for an historic and mathematical treatment of these methods). Similarly, the district magnitudes c = (c 1 , c 2 , . . . , c m ) are computed based on, e.g., district population counts. Note that we have i r i = j c j = h. In biproportional apportionment the problem is to assign seat numbers to each party within each district such that the party (row-sum) requirements and district (column-sum) requirements are fulfilled. Formally, we should compute an n × m matrix X = (x ij ) of nonnegative integers where x i * := j x ij = r i and x * j := i x ij = c j for all i, j. This should furthermore be done in such a way that the seat numbers x ij are proportional to the vote counts p ij for all i, j. Variants and properties of biproportional apportionment were studied by Anthonisse [1], De Meur et al [31] and Gassner [22, 23] . Gassner showed that in the problem with pre-specified marginals, it is impossible to guarantee proportionality at the local level (e.g., within districts) unless there are at most two parties and at most two districts. A biproportional apportionment can be computed using a divisor-based (or multiplier-based) method as suggested by Balinski and Demange [6]. This method works by computing row multipliers λ i and column multipliers µ j , such that and such that the row-and column-sum requirements are fulfilled; the function [·] rounds a fractional number q to either q or q . Balinski and Demange [6] showed that divisor-based methods have several important and unique properties that make them well-suited for use in electoral systems. The Zürich Canton Parliament recently chose to use divisor-based biproportional apportionment for the distribution of its seats; consequently, the same apportionment system is now used for the Zürich City Council [36] . In Table 1 the vote numbers and resulting seat numbers for the Zürich City Council election on February 12, 2006 are shown: This election was the first time ever that a divisor-based biproportional apportionment method was used for distributing parliament seats. Applications to other parliaments, including comparisons to alternative apportionment methods, are studied in [3, 4, 8, 11, 28, 42] . A software package named BAZI is available for computing vector and matrix apportionments using divisor-based methods [29, 34] . The purpose of our paper is to evaluate the applicability of divisor-based biproportional apportionment in electoral systems. Our benchmark study is based on simulation: Given a set of actual election results (vote counts), we randomly generate realistic (but artificial) election data, and use biproportional apportionment to assign the seats. This allows us to estimate both the expected behavior of the tested algorithms, and the average quality of seat distributions obtained from divisor-based biproportional apportionment. Furthermore, this makes it possible to estimate how often extreme situations appear -such as the occurrence of ties. The use of extensive computer simulation for benchmarking electoral systems is still in its infancy. Benoit [10] examines some of the classical electoral formula using simulated election results generated from actual vote distributions. Christensen [12] describes a simulation approach for comparing six different single-winner voting procedures by using five different criteria. Fragnelli et al [17, 18] argue strongly for the use of simulation for the choice of an electoral system,