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

Maier, Sebastian; Zachariassen, Petur; Zachariasen, Martin

doi:10.1287/mnsc.1090.1118

Cited by 39 publications

(15 citation statements)

References 32 publications

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“…In particular, in any election there is the possibility of ties: consider how any system, whether FPTP or proportional, fairly allocates seats in an election where all parties earn the same number of votes in every constituency. Maier et al (2010) analysed realistic data for districts with multiple representatives and found no instances of non-uniqueness. We revisit uniqueness in the context of network flow later in the paper and find that for single seat constituencies a judicious choice of objective function seems to prevent a threat of multiple solutions.…”

Section: Corollary 1 So Long As No Party Is Awarded More Seats Than Imentioning

confidence: 99%

“…In our experiments, we have found that only one or two such steps are needed. (Maier et al 2010) describe a number of algorithms for the integer problem including a discrete version of (1) based on alternating iterative vector apportionment of the multipliers to achieve the desired row and column sums. We have adapted their algorithm to take advantage of the fact that since UK constituencies each elect a single MP x i j ∈ {0, 1} which removes the guesswork needed to find initial estimates of the multipliers at each step.…”

Section: Biproportional Roundingmentioning

confidence: 99%

“…Finding the closest fit is known as the biproportional apportionment problem. It was first proposed as a system for proportional representation in (Balinski and Demange 1989) and has since been adopted successfully in a number of legislatures (Balinski and Pukelsheim 2006;Maier et al 2010). It is possible to apply biproportional rounding to single seat constituencies.…”

Section: Introductionmentioning

confidence: 99%

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Network models and biproportional rounding for fair seat allocations in the UK elections

Akartunalı

Knight

2016

Ann Oper Res

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Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. Biproportional rounding, which is an instance of the network flow problem, is used in some European countries with multi-seat constituencies. We discuss its application to single seat constituencies and the inevitable consequence that seats are allocated to candidates with little local support. However, we show that variants can be selected, such as regional apportionment, to mitigate this problem. In particular, we introduce a parameter based family of methods, which we call balanced majority voting, that can be tuned to meet the public's demand for local and global "fairness". Using data from the 2010 to 2015 UK general elections, we study a variety of network models and implementations of biproportional rounding, and address conditions of existence and uniqueness.

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Section: Corollary 1 So Long As No Party Is Awarded More Seats Than Imentioning

confidence: 99%

Section: Biproportional Roundingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Network models and biproportional rounding for fair seat allocations in the UK elections

Akartunalı

Knight

2016

Ann Oper Res

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“…Balinski & Pukelsheim (2006) conjecture that these pathological instances only appear in case of especially complicated ties. Maier (2008) and Maier et al (2010) discuss the interplay between the AS-algorithm and the Tie&Transfer-algorithm. Intensive simulations suggest that the hybrid algorithm, which starts with alternate scaling and then switches to the Tie&Transfer-algorithm, is faster than the simple algorithms.…”

Section: Literature Surveymentioning

confidence: 99%

“…Simulations by Maier et al (2010) suggest that the set of biproportional apportionments can be determined as fast as possible if the procedure is initialized by the AS-algorithm and switches to the Tie&Transfer-algorithm as soon as the L 1 -error remains unchanged for two steps.…”

Section: As-tt-combinationmentioning

confidence: 99%

Alternate Scaling algorithm for biproportional divisor methods

Oelbermann

2016

Mathematical Social Sciences

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In parliamentary elections biproportional divisor methods 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.

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Network flow methods for electoral systems

et al. 2011

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Researchers in the area of electoral systems have recently turned their attention to network flow techniques with the aim to resolve certain practically relevant problems arising in this area. The aim of the present paper is review some of this work, showing the applicability of these techniques even to problems of a very different nature.Major emphasis will be placed on "biproportional apportionment", a problem that frequently arises in proportional electoral systems, but which in some countries is still ill-solved, or not dealt with rigorously, notwithstanding the availability of several sound solution procedures and their concrete application in some real-life elections. Beside biproportional apportionment, we shall discuss applications of network flows to problems such as vote transitions and political districting. Finally, we address the so-called "Give-up Problem", which arises in the current elections for the Italian Parliament. It is related to the possible assignment of seats to multiple winners of a given party. Based on the results and techniques presented in this paper, it is fair to state that network flow models and algorithms are indeed very flexible and effective tools for the analysis and the design of contemporary electoral systems.

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

Cited by 39 publications

References 32 publications

Network models and biproportional rounding for fair seat allocations in the UK elections

Network models and biproportional rounding for fair seat allocations in the UK elections

Alternate Scaling algorithm for biproportional divisor methods

Network flow methods for electoral systems

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