Tackling the Bundestag growth by introducing fraction-valued votes

Currently, only China has a parliament larger than the German Bundestag, which continues to grow due to the increasing number of overhang mandates. In 2016, Norbert Lammert, then president of the Bundestag, proposed to restrict it to 630 members by allocating mandates according to quotas for each of the German states (Länder), which should be proportional to their population. This idea found no approval among the German parties, neither large nor small [Finthammer 2018]. Only in October 2019, after predictions that the next Bundestag could exceed 800 seats, did some 100 German experts in constitutional law write an open letter suggesting to constrain its size by reducing the number of effective constituencies, and the Bundestag vice-president, Thomas Oppermann, called for such a reform without delay. These and other proposals require a profound change in the existing election system. But a mathematical solution to the problem does not require such changes and is much simpler. We can prevent unfettered growth of the Bundestag - caused by allotting too many direct mandates to parties that received too few second votes - by replacing the principle of "one man, one vote" with a new concept: fraction-valued votes for Bundestag members. Such a practice could make overhang mandates unnecessary and the basic 598 Bundestag seats sufficient under all circumstances. For this purpose, the members of the overrepresented parties (because they receive too many direct mandates) should have vote power < 1 and the members of other parties should have vote power > 1. We explain the vote power adjustments using the example of the 2017 Bundestag.