Open Access

Udo Schwingenschlögl

• Because in democratic systems the electoral outcome decides on the line of future policy, the process of voting is of great importance for society. In general, an election consists of two parts, both influencing its result. In the first step, each voter gives his vote to one of the parties participating in the election. The numbers of votes in favor of the competing parties then give rise to vote proportions, which specify the share of voters supporting a party. In the second step, almost continuous vote proportions have to be translated into integer numbers of seats in the parliament. Translating electoral votes into specific seat allocations, the process of apportionment, unavoidably influences the final distribution of power, because in general it involves some kind of adjustment of the fractional seats that would arise if literal calculation were possible. As a consequence, it is an important issue in proportional representation systems to measure the effects of this adjustmentBecause in democratic systems the electoral outcome decides on the line of future policy, the process of voting is of great importance for society. In general, an election consists of two parts, both influencing its result. In the first step, each voter gives his vote to one of the parties participating in the election. The numbers of votes in favor of the competing parties then give rise to vote proportions, which specify the share of voters supporting a party. In the second step, almost continuous vote proportions have to be translated into integer numbers of seats in the parliament. Translating electoral votes into specific seat allocations, the process of apportionment, unavoidably influences the final distribution of power, because in general it involves some kind of adjustment of the fractional seats that would arise if literal calculation were possible. As a consequence, it is an important issue in proportional representation systems to measure the effects of this adjustment process, in order to judge which apportionment method is most suitable for application. This thesis concentrates on some of the most popular apportionment methods: The quota method of greatest remainders and the stationary divisor methods. It is investigated whether these apportionment methods, on average, treat smaller and larger parties equally or allow a systematic advantage in either direction. For measuring the effect of the adjustment process, the concept of seat biases has been introduced. Seat biases are defined as averages of the difference between the seats actually apportioned to the competing parties and their ideal shares of seats. Of course, apportionment methods should result in vanishing seat biases for legal reasons. Assuming repeated application of these methods, it is possible to determine the seat biases affecting the various parties. A geometric-combinatorial approach to the calculation of seat biases is introduced, which turns out to be highly useful in order to evaluate this expectation. It is based on a combination of knowledge about the geometry of sets of vote proportions leading to a specific seat allocation and of a combinatorial method of accounting for all possible seat allocations. The political character of the problem of a violated proportionality calls for quantitative seat bias results, which become accessible in a rigorous fashion by means of the geometric-combinatorial approach.…
• Im Kontext einer Verhältniswahl müssen die Stimmenanteile der einzelnen Parteien in Mandate umgerechnet werden. Zu diesem Zweck steht eine Vielzahl verschiedener Zuteilungsmethoden zur Auswahl, die sich im Hinblick auf die aus der Mandatszuteilung resultierende Verzerrung der ursprünglichen Stimmenanteile unterscheiden. In der vorliegenden Arbeit werden diese Verzerrungen unter Verwendung eines geometrisch-kombinatorischen Ansatzes untersucht und quantitativ bestimmt.