Candidate elected party vote elected party vote

Part 5 of 5:

A New MMP Method (Part 2)

This paper is the fifth part of a series of papers that can be downloaded here:

Contents

An MMP scheme is an election method where each voter has two votes: a district vote and a party vote. Let’s say that the standard size of the parliament is 100 seats. Then the electorate is divided into districts; a certain number of seats (say 60 seats) is allocated to the districts; with the district vote, the voters choose the district winners. The party vote is used to determine how many seats each party deserves when all 100 seats are allocated by proportional representation on the national level. When a party wins X district seats and deserves Y > X seats according to its number of party votes, then this party gets, in addition to its district seats, Y - X extra seats. When a party wins X district seats and deserves Y < X seats according to its number of party votes, then this party keeps all these district seats, so that the total size of the parliament is increased by X - Y (overhang seats).

For the sake of simplicity, we ignored, in the fourth part of this series of papers, the most serious strategic problem of MMP methods: Under MMP methods, it is a useful strategy for a party A to run two lists (A1 and A2) and to tell its supporters to give their district votes to the candidates of list A1 and their party votes to list A2. When its supporters vote as they are told, then list A1 receives significantly more district seats than it deserves seats according to its number of party votes. These extra seats for list A1 are extra seats for party A, because list A2 already receives as many seats as party A deserves according to the proportional share of its number of supporters.

If the maximum number of compensation seats was not limited, then the total size of the Assembly would have been around 730 seats ( = 56 seats · 100% / 7.67% ).

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Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

2. The District Vote

Table 2.1.1: The 12 Berlin boroughs

We recommend that the districts for the elections to the Berlin House of Representatives should be the 12 Berlin boroughs. When the Hill-Huntington method is being used to allocate the 115 district seats to the 12 districts, then we get two 8-seat districts (Friedrichshain-Kreuzberg, Spandau), five 9-seat districts (Mitte, Neukölln, Treptow-Köpenick, Lichtenberg, Reinickendorf), three 10-seat districts (Charlottenburg-Wilmersdorf, Steglitz-Zehlendorf, Marzahn-Hellersdorf), one 11-seat district (Tempelhof-Schöneberg), and one 13-seat district (Pankow).

The individual voter ranks the candidates in order of preference. The individual voter may ...

... give the same preference to more than one candidate.

... skip preferences. However, skipping some preferences does not have any impact on the result of the elections, since the result of the elections depends only on the order in which the individual voters ranks the candidates and not on the absolute preferences of the individual voters.

... give preferences to parties. When a given voter gives a preference to a party, then this means that each candidate of this party gets this preference unless this voter explicitly gives a different preference to this candidate.

4. Junge-Reyer (SPD, candidate 01.001) → elected (district vote) 5. Ratzmann (B‘90G, candidate 04.001) → elected (district vote)

6. İzgin (Left, candidate 03.005)
7. Klotz (B‘90G, candidate 04.003) 8. Samuray (CDU, candidate 02.004)

05: Free Democratic Party of Germany (FDP)

06: The Republicans (REP)

However, when the district winners are chosen by an STV method, then it is not immediately clear which voter voted for which district winner. Therefore, at first, we have to calculate, for each district separately, an allocation of the voters to the district winners.

Suppose Aelected⊆A is the set of district winners. Suppose M is the number of district winners.

Suppose λij∈— is that share of voter i∈V that is allocated to candidate j∈Aelected. Then λij must satisfy at least the following three conditions:

Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

Suppose ∀i∈V∀j∈Aelected: ηij∈Õ is the number of candidates k∈Aelected with kêij. In other words: ηij∈Õ is the number of candidates who are strictly preferred by voter i to candidate j.

by minimizing the total number of votes that are transferred to a candidate j although the voter i strictly prefers another candidate k to candidate j.

That means:
Suppose also µij satisfies (2.4.1) -- (2.4.3). Suppose the voters are sorted in such a manner τ that

Condition #2:
Suppose voter i is indifferent between candidate m and candidate n.

3. The Party Vote

3.1. The Party Ballot

Suppose pi is the number of voters whose district votes are allocated to a district winner of party i and who do not cast a valid party vote.

Suppose qij is the number of voters whose district votes are allocated to a district winner of party i and who give their party vote to party j.

is the effective number of party votes for party i. In the beginning, zi : = 0 for all parties i.

Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

a deserves according to its effective number of party votes or (2) za = 1.

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Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

We take party a with minimum ra.

We increase za continuously. Simultaneously, we adjust εi for all parties i continuously. Simultaneously, we adjust continuously the 131 largest quotients that we get when we divide the effective numbers of party votes by 0.75, 1.5, 2.5, 3.5, 4.5, 5.5, etc..

(3.2.2) zi = 1 for at least one party i.

(3.2.3) The set of parties with minimum ri has increased.

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Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

ri : = εi / ( Di - 0.5 ) for each party i with Di > 1.
ri : = εi / 0.75 for each party i with Di = 1.
ri : = ∞ for each party i with Di = 0.

Y : = miniri.

T : = maxiti.

U : = maxi { Di | ti = T }.

(b) Σjsj is odd.

(c) T ≤ Y.

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Markus Schulze, “Part 5 of 5: A New MMP Method (Part 2)”

We recommend that this allocation procedure should have the following

properties:

Therefore, we propose the following method to allocate the seats of party i

to this party’s district organizations:

of party i in district j.

Stage 1:

i > 0.

tj with sj i / 0.75 for each district organization j of party i i : = ε j

and Dj i = U. If there is more than one district

organization with tj i = T and D j i = U, then we decide

3.4. The Party Vote Winners

Now we calculate the remaining places of the proportional rankings of the districts. In each district j, we ignore all the candidates of parties i with Sj i= 0.

1. Mutlu (B‘90G, candidate 04.002) 2. Fischer (SPD, candidate 01.006) 3. Reinauer (Left, candidate 03.009)

→ elected (district vote) → elected (district vote) → elected (district vote)

12. Wolf (Left, candidate 03.002) 13. Altug (B‘90G, candidate 04.008) 14. Bayram (SPD, candidate 01.005)

→ elected (party vote)
→ elected (party vote)

When a seat gets vacant, then this seat goes to that candidate of this party who is ranked highest in the proportional ranking of this district. If the list of candidates of this party is exhausted or if this seat was the seat of an independent candidate, then this seat stays vacant.

Example: The seat of Klotz gets vacant. Then this seat goes to Kosche.

→ elected (party vote) → elected (party vote)

→ elected (successor of Klotz)