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Beschreibung:
We critically discuss the Jefferson/D'Hondt and Webster/Sainte-Laguë methods, which are used to allocate parliament seats to parties in the mixed-member proportional representation systems in Germany, New Zealand, Bolivia, South Africa, South Korea, Scotland and Wales, as well as in the European Parliament. The task is as follows: (1) the parliament must be of a certain size or slightly larger than that, (2) the party factions must include all direct mandate holders elected in constituencies and (3) the faction ratio should reflect, with a certain accuracy, the votes received by the parties across the country. We show that discrete optimization techniques result in better and more accurate apportionments. In addition, we consider adjustment vote weights defined within the optimization approach and show that they can give a general consistent solution to the apportionment problem. All of these are illustrated using the example of the 2021 German Bundestag elections.