Type of mixed electoral system
Mixed-member proportional representation (MMP or MMPR) is a type of representation provided by some mixed electoral systems which combine local winner-take-all elections with a compensatory tier with party lists, in a way that produces proportional representation overall. Like proportional representation, MMP is not a single system, but a principle and goal of several similar systems. Some systems designed to achieve proportionality are still called mixed-member proportional, even if they generally fall short of full proportionality. In this case, they provide semi-proportional representation.
In typical MMP systems, voters get two votes: one to decide the representative for their single-seat constituency, and one for a political party, but some countries use single vote variants. Seats in the legislature are filled first by the successful constituency candidates, and second, by party candidates based on the percentage of nationwide or region-wide votes that each party received. The constituency representatives are usually elected using first-past-the-post voting (FPTP). The nationwide or regional party representatives are, in most jurisdictions, drawn from published party lists, similar to party-list proportional representation. To gain a nationwide representative, parties may be required to achieve a minimum number of constituency seats, a minimum percentage of the nationwide party vote, or both.
MMP differs from mixed-member majoritarian representation (often achieved by parallel voting) in that the nationwide seats are allocated to political parties in a compensatory manner in order to achieve proportional election results across all seats (not just the additional seats). Under MMP, two parties that each receive 25% of the votes end up with about 25% of the seats, even if one party wins more constituency seats than the other. Depending on the exact system implemented in a country and the results of a particular election, the proportionality of an election may vary. Overhang seats may reduce the proportionality of the system, although this can be compensated for by allocating additional party list seats to cover any proportionality gap.
The specific system of New Zealand for electing its parliament is called MMP, while in other countries similar systems are known under other names.
Other names.
The seat linkage compensatory mixed system often referred to as MMP originates in Germany, and was later adopted with modifications under the name of MMP in New Zealand. In Germany, where it was differentiated from a different compensatory mixed system it was always known as "personalized proportional representation (PPR)" (). Since the variants used in Germany almost always produce very proportional results, the proportionality is emphasized over the mixed nature of the electoral system, and it is essentially considered a localized or personalized form of PR, used instead of conventional open-list systems. The new modified federal election system does not allow overhang seats at all, therefore not all local districts are guaranteed to elect the plurality winner. In German, this localized list system now shares the name of PPR with the mixed systems still used in the federal states of Germany that are referred to as MMP in English. In English, due to this change, the system is no longer considered to be MMP in the sense of a mixed member system combining proportional and majoritarian representation, but it would be a personalized/localized version of PR. As it retains the individual candidate vote in a clearly distinct fashion from open-list systems, it may still be considered mixed-member proportional in the sense of a proportional system having two kinds of MP: one (may be) elected by personal (candidate) votes, one elected by (closed list) votes. 
Previously, the federal elections used a flexible number of additional compensatory seats, also known as leveling seats, which essentially guaranteed mixed-member proportional representation even with extremely disproportional constituency results, but dramatically increased the size of the Bundestag. This meant that it was potentially the most proportional MMP system used after the one in New Zealand, where only overhang seats are added back as list seats, which resulted in minor flexibility of the parliament size.
In the Canadian province of Quebec, where an MMP model was studied in 2007, it is called the "compensatory mixed-member" voting system ( or SMAC). In the United Kingdom the sometimes less proportional implementation of MMP used in Scotland and the London Assembly is referred to as the additional member system. In South Africa, MMP is generally referred to as a "mixed-system". The Scandinavian countries have a long history of using both multi-member districts (members elected through party-list PR) and nationally-based compensatory top-up seats using the same method as MMP, however because the local MPs are also elected using PR, these systems are not usually considered MMP as they are not mixed systems. 
As especially mixed electoral systems can be quite different, sometimes there is no consensus on their classification as mixed-member proportional, mixed majoritarian or something between the two. These cases include partially or conditionally compensatory systems such as those of Hungary, Mexico and South Korea, which are typically said to be supermixed systems or partially compensatory systems, but sometimes inaccurately referred to as MMP even though they are highly disproportional.
Procedures.
In MMP, the voter casts two votes: one for a constituency representative and one for a party. In the original variant used in Germany, citizens gave only one vote, so that voting for a representative automatically meant also voting for the representative's party, which is still used in some MMP elections today and is more robust against tactical voting than typical two-vote versions. Most of Germany changed to the two-vote variant to make local members of parliament (MPs) more personally accountable. Voters can thus vote for the local person they prefer for local MP without regard for party affiliation, since the partisan make-up of the legislature is determined only by the party vote. In the 2017 New Zealand election, 27.33% of voters split their vote (voted for a local candidate of a different party than their party vote) compared to 31.64% in 2014.
In each constituency, the representative is by default chosen using a single winner method (though this is not strictly necessary), typically first-past-the-post: that is, the candidate with the most votes (plurality) wins.
Most systems used closed party lists to elect the non-constituency MPs (also called list MPs). In most jurisdictions, candidates may stand for both a constituency and on a party list (referred to in New Zealand as "dual candidacy"). In Wales between 2006 and 2014 dual candidacy was banned, i.e. candidates were restricted to contend either for a constituency or for a party list, but not both. If a candidate is on the party list, but wins a constituency seat, they do not receive two seats; they are instead crossed off the party list and the party seat goes to the next candidate down.
In Bavaria, the second vote is not simply for the party but for one of the candidates on the party's regional list: Bavaria uses seven regions for this purpose. A regional open-list method was recommended for the United Kingdom by the Jenkins Commission (where it is known as AMS) and for Canada by the Law Commission of Canada; neither recommendation was ever implemented.
In contrast, the open-list method of MMP was chosen in November 2016 by voters in the 2016 Prince Edward Island electoral reform referendum.
In Baden-Württemberg, there were no closed lists prior to 2022; they used the "best near-winner" method in a four-region model, where the regional members are the local candidates of the under-represented party in that region who received the most votes in their local constituency without being elected in it (Zweitmandat, literally "second mandate").
Apportionment methods.
At the regional or national level (i.e. above the constituency level) several different calculation methods have been used, but the basic characteristic of the MMP is that the total number of seats in the assembly, including the single-member seats and not only the party-list ones, are allocated to parties proportionally to the number of votes the party received in the party portion of the ballot. This can be done by different apportionment methods: such as the D'Hondt method or the Sainte-Laguë method. Subtracted from each party's allocation is the number of constituency seats that party won, so that the additional seats are compensatory (top-up).
Dealing with overhang seats.
If a party wins more FPTP seats than the proportional quota received by the party-list vote, these surplus seats are called overhang seats ( in German), which may be an obstacle to achieving full proportionality.
When a party wins more constituency seats than it would be entitled to from its proportion of (party list) votes, most systems allow for these overhang seats to be kept by those candidates who earned it in the constituency elections. A counter-example would be the In Germany's "Bundestag", where constituency winners may not always keep their seats since the latest modification of the electoral law. In an MMP variant used in Romania in the 2008 and 2012 legislative elections, where constituency seats could only be earned by the winning candidate if they also achieved an absolute majority in their district, thereby eliminating overhang seats.
In New Zealand House of Representatives, all members elected for constituencies keep their seats. For example, in the 2008 New Zealand general election the Māori Party won 2.4% of the party vote, which would entitle them to 3 seats in the House, but won 5 constituency seats, leaving an overhang of 2 seats, which resulted in a 122-member house. If the party vote for the Māori Party had been more in proportion with the constituency seats won, there would have been a normal 120-member house.
To combat disproportionalities caused by overhang seats, in most German states leveling seats ("Ausgleichsmandate" in German) are added to compensate for the overhang seats and thereby achieve complete proportionality. For example, the provincial parliament ("Landtag") of North Rhine Westphalia has, instead of the usual 50% compensatory seats, only 29% unless more are needed to balance overhangs. If a party wins more local seats than its proportion of the total vote justifies, the size of the "Landtag" increases so that the total outcome is fully proportional to the votes, with other parties receiving additional list seats to achieve proportionality. The leveling seats are added to the normal number of seats for the duration of the electoral period. In the German state of Bavaria, the constituency vote and party vote are combined to determine the distribution of seats.
Scotland uses a modified variant of MMP known as the additional member system where due to the nature of the calculations used to distribute the regional list seats, overhang seats are not possible; the list allocation works like a mixed-member majoritarian system, but in using the d'Hondt method's divisors to find the averages for the allocation, the first divisor for each party takes into account the number of constituency seats won by the party; i.e. a party that won 7 constituency seats would start with a divisor of 8 (7 seats + 1 per the method's divisor formula) instead of 1. The resulting table would then give 7 seats for Scotland and 4 seats for Wales to the parties possessing the highest averages on the table, although both devolved parliaments do not use a table, instead using a sequential method. The compensatory effect characteristic of MMP is in the fact that a party that won constituency seats would have lower averages on the table than it would if the election used MMM. Because of no provision for overhang seats, there have been cases where a party ended up with fewer total seats than its proportional entitlement. This occurred, for example, in the elections in the South East Wales electoral region in both 2007 (Welsh Conservatives under-represented) and in 2016 (Welsh Labour over-represented, Plaid Cymru under-represented). Labour has also been over-represented on this basis in every election in the South Wales West region, and every election in the South Wales Central region apart from the 2003 election. This situation arises because Labour has continued to hold the overwhelming majority of constituency seats in these regions, and only around one-third of the total number of seats are available for distribution as additional regional seats.
Threshold.
As in numerous proportional systems, in order to be eligible for list seats in many MMP models, a party must earn at least a certain percentage of the total party vote, or no candidates will be elected from the party list. Candidates having won a constituency will still have won their seat.