How are levelling seats allocated in Scandinavian elections?

Question

In Scandinavian countries such as Denmark and Sweden, their parliaments tend to be comprised of district seats (allocated based on vote count within the district) and levelling seats (allocated based on vote count nationwide).

It's not clear to me how they allocate levelling seats to political parties when they are already using open-list proportional representation. From what I understand, there are two possible ways of doing this:

1. Each party makes a closed-list before the election, and they just run it down based on how much adjustment seats they get after the votes are counted.
2. After the district seats are counted, each party combines all the lists from every district then forms a new list. They then rank the un-elected candidates based on the vote they received, and continue to run down this combined list based on how much adjustment seats they get.

How does it actually work?

Answer 1

The system for using lists to fill the leveling seats (or "adjustment seats") varies per country. This answer summarises information from this document (PDF) published by the European Committee for Democracy Through Law in 2015.

In particular, we examine the last column of the table beginning from page 2.

Denmark (begins page 32)

• Candidates who have met a threshold for direct election are assigned a seat.

• where a party has insufficient candidates directly elected to meet the proportion of seats earned, the remaining seats are allocated to candidates in order from a list submitted by the party before the election, for that region specifically.

• Where a party has insufficient candidates elected in the regions to meet the proportion of seats earned, they are awarded a number of leveling seats to make up the difference.

• Leveling seats are allocated to candidates in order from a list submitted by the party before the election, for the leveling seats specifically.

Norway (begins page 111)

• Parties submit a list for the leveling seats purpose before the election, which is included in the voting papers.

• Voters may propose a new list order (including rejection of candidates on the party's list, but not additions)

• If a simple majority of voters who vote for that party propose the same new ordering, that ordering is adopted. Otherwise, where the party has insufficient candidates elected in the regions to meet the proportion of seats earned, they are awarded a number of leveling seats to make up the difference.

• Leveling seats are allocated to candidates in order from the list submitted by the party before the election, for the leveling seats specifically, unless voters provide a new list by consensus which is applied instead.

Answer 2

https://en.wikipedia.org/wiki/Leveling_seat seems to explain it

TLDR; is Denmark has 179 seats, 4 of those are for north atlantic representation, 2 faeroe island, and 2 for greenland. remaining 175 is then what your talking about. of those 135 are district seats, and 40 are leveling seats, or as you call them adjustment seats.

As I read it, even if districts are given seats proportional to population. Small districts only getting 1 or 2 representatives, would almost be guarantied to always go to the large parties. regardless of the actual vote. So Adjustment seats are to give those parties that got perhaps 10% of the votes in primarily small districts a representation too. Where they would otherwise not be represented.

Answer 3

39 seats (349 in total) in the Swedish Riksdagen is utjämningsmandat (leveling seats)

It is an modified Sainte-Laguë method.

All parties can leave notifications of their candidates with valmyndigheten (state election administration) pre-election (locked/closed candidate lists.) If a party doesn't create these lists but get votes (unlocked list), the voter can nominate by name some person.

See for example wiki on Webster-Sainte Laguë