[A ROUGH FIRST DRAFT. Please comment.]
This system is used in Scottish parliamentary elections [and also in elections for Greater London Assembly, I think].
The Electoral Commission website says:
Constituency candidates are elected using the first-past-the-post. Once the constituency votes are counted the candidate with the most votes is declared elected.
The regional votes are then counted. The Regional Returning Officer add up the total regional votes from all of the constituencies in their region cast for parties or individual candidates. They will also add up the total number of constituency seats won within the region by each registered party.[1]
The number of votes cast in the regional ballot for each registered party or individual candidate is divided by the number of constituency seats gained, plus one. After that calculation is done, the party with the highest figure gets the first regional seat. To allocate the second to seventh additional seats the calculation is redone, but each time any additional seats gained are added into the calculation.
This method of calculation is known as the modified d’Hondt system.[2]
There is much support for the adoption of AMS, but also some objections.
A few are opposed to the introduction of multi-seat election, favouring the direct connection between an MP and their constituents. But AMS partially addresses this by having both its constituency and its regional components.
Some will object on the grounds that under AMS it is parties rather than the voters decide who their MPs will be. Which AMS does as a simple way of ensuring “proportionality”. In any case, it is not so different to what happens now, particularly in safe seats.
AMS also makes success unreasonably difficult for small parties and for independent candidates.
Perhaps more significantly, a serious short-term problem is that AMS is extremely unlikely to be feasible before the next UK election. It will take time considerable time to formulate and enact the necessary legislation. To prepare for AMS then requires boundary changes, if only grouping of constituencies into pairs[1] and into new regions. And then further time is needed to allow political parties and local administrators to reorganise to the new structure. Early implementation seems distinctly infeasible.
So if we pursue AMS to the exclusion of any more immediate solution then you should assume that we shall have to endure at least one more general election under FPTP. Not a cheerful prospect.
[1] We assume that seats are 50:50 constituency/regional. So if UK is to have the same number of MPs then the constituencies have to be twice as big as with pure FPTP.