Majority bonus system

The majority bonus system (MBS) is a form of semi-proportional representation used in some European countries. Its feature is a majority bonus which gives extra seats or representation in an elected body to the party or to the joined parties with the most votes with the aim of providing government stability.
It is used in San Marino, and Italy from 2006 to 2013. In Argentina it is used in the Chamber of Deputies of the Province of Santa Fe, Chubut, and Entre Ríos. Greece used the system before abolishing it in 2016 but the law will not take effect until the second election after it was passed, so the majority bonus system was used in the 2019 election before being abolished officially.
History.
Benito Mussolini was the first politician to enact a law to give automatic seats to the winning party and ensured his victory in the Italian election of 1924. A modified version of the system was reintroduced for the 1953 general election, in which any parliamentary coalition winning an absolute majority of votes would be awarded two-thirds of the seats in Parliament. The Christian Democracy-led coalition fell narrowly short of this majority in the election, and the system was abolished before the 1958 election. The majority bonus system was used in Italian local elections in the 1950s and was reintroduced for local elections in 1993 and national ones in 2006 to replace the "scorporo" mixed system. In the Italian election of 2013, the Democratic Party won 292 seats in the House using its 8,644,523 votes and so needed 29,604 preferences to obtain a seat. Its major opponent, The People of Freedom, won 97 seats with 7,332,972 votes and so needed 75,597 votes for a single seat. Effectively, the system in use in Italy from 2006 until 2013, which assigned the jackpot regardless of the percentage of vote achieved by the largest party, was judged as unconstitutional by the Italian Constitutional Court. After a proposed modification involving a run-off vote (between the top two alliances) was also struck down by the court, parallel voting was adopted for the Italian election of 2018.
The majority bonus system was adopted by other European countries, especially Greece in 2004 and San Marino at the national level, and France for its regional and municipal elections.
Mechanism.
It can be based on any form of mechanism used in party-list proportional representation, but a D'Hondt method is most likely, as it will rank seats in an exact order of vote share.
Basically, there are two different forms of majority bonus systems, with clearly different political results:
The jackpot system assures a fixed (minimum) number of seats to the winner, while the bonus system adds a fixed number of seats.