MMP-Elections : Equal Influence and Controlled Assembly Size

In an MMP election (Mixed Member Proportional) of a legislature, a QP-ballot supports party Q in a single-seat constituency and a list of candidates from P. With ꙍ(j) constituency seats won and list support in z(Pj) ballots, party Pj wins α(j) list seats, so that ꙍ(j)+α(j) becomes proportional to z(Pj).The pivotal party, Pj*, has the highest of all ratios ꙍ(j)/z(Pj). Proportionality implies, for all Pj passing some threshold, that [ꙍ(j)+α(j)]/z(Pj) ≥ ꙍ(j*)/z(Pj*).In the smallest proportional assembly, all ≥ are equalities and α(j*)=0.The pivotal party’s list support, z(Pj*), is naturally volatile. An election with α(j*)=0 tells that z(Pj*) list votes were wasted, and many voters learn it. Thus, between Bundestag elections 2017 and 2021, z(Pj*) dropped significantly. The smallest possible size of a proportional assembly rose from 709 to 794 seats, while the legal norm is 598.But an ad-hoc law of 2020 abandoned the proportionality rule, shrinking the assembly from 794 to 736 seats.ꙍ(j) measures and records the success of party Pj in the single-seat tallies; it also records how much α(j) is reduced by Pj’s constituency success. The paper compares this”traditional accounting” and“faithful accounting”: The latter records a QiPj-ballot, with Qi as constituency winner, with a tiny seat fraction that reduces α(j). Traditional accounting treats party Pj as a basic entity. Faithful accounting replaces it by the set Λ(Pj) of voters with list vote for Pj.This is a paradigm shift: Traditional accounting works even if constituency votes and list votes are collected in separate ballot boxes. But in faithful accounting, each ballot’s combination of Qi and Pj is essential.Main results: The change from traditional to faithful accounting brings the assembly size under control. A large inequality in voters influence is substantially reduced