I wanted more details on how this works. For those interested, I found an English pdf describing the full system [1]. The interesting part is Article 110, which discusses how the adjustment seats are allocated. Here is my best summary:
1. Using D'Hondt [2] on every party's national vote share, determine which party should be given the next seat.
2. For every constituency which has adjustment seats available, calculate the D'Hondt quotient of the first candidate in that party who has not already been elected using the constituency vote share. So if a party received V votes in a constituency and two party members were already elected from this constituency, their quotient would be V/3.
3. Elect the candidate with the highest quotient to fill an adjustment seat for their constituency.
4. Repeat until all adjustment seats have been given away.
There's arguably a step 0 here, which is determining how many constituency and adjustment seats every constituency gets, and this is done before the election is held. This is described in Article 10. It's pretty bad. First, the adjustment seats are hard-coded. Second, unlike the US where we reapportion after every census, Iceland appears to only reapportion the constituency seats when the constitution demands they do it. This happens when there are twice as many voters per seat in one constituency compared to another. Furthermore, they only adjust as few seats as possible to get back under this limit rather than actually recalculate a fair apportionment. I'm not sure what the logic of this was, maybe to minimize how often the number of seats in each place is changing? Either way, in the 2021 election this resulted in one constituency with 199% as many voters per seat as another and no changes were made [3].
1. Using D'Hondt [2] on every party's national vote share, determine which party should be given the next seat. 2. For every constituency which has adjustment seats available, calculate the D'Hondt quotient of the first candidate in that party who has not already been elected using the constituency vote share. So if a party received V votes in a constituency and two party members were already elected from this constituency, their quotient would be V/3. 3. Elect the candidate with the highest quotient to fill an adjustment seat for their constituency. 4. Repeat until all adjustment seats have been given away.
There's arguably a step 0 here, which is determining how many constituency and adjustment seats every constituency gets, and this is done before the election is held. This is described in Article 10. It's pretty bad. First, the adjustment seats are hard-coded. Second, unlike the US where we reapportion after every census, Iceland appears to only reapportion the constituency seats when the constitution demands they do it. This happens when there are twice as many voters per seat in one constituency compared to another. Furthermore, they only adjust as few seats as possible to get back under this limit rather than actually recalculate a fair apportionment. I'm not sure what the logic of this was, maybe to minimize how often the number of seats in each place is changing? Either way, in the 2021 election this resulted in one constituency with 199% as many voters per seat as another and no changes were made [3].
[1]: https://www.stjornarradid.is/library/03-Verkefni/Kosningar/K...
[2]: https://en.wikipedia.org/wiki/D%27Hondt_method
[3]: https://en.wikipedia.org/wiki/2021_Icelandic_parliamentary_e...