Majority systems ensure that the winning candidate will achieve more than 50% of the vote in a constituency, though they don't guarantee that - in a country as a whole - the results of general elections reflect the proportion of votes cast for each party.
We usually distinguish three best known systems:
1. the Alternative Vote system
Under the AV, as it's often called, the voters rank all the candidates (whose name appear on the ballot) in order of preference. As in FPTP, candidates stand for election in constituency and one MP is elected in each area.
If any candidate receives more than 50% of first preferences in the initial ballot, then that candidate is elected.
If, however, no one gains majority of votes, the candidate with the lowest number of first preferences is eliminated and his second preferences are redistributed to the other candidates. This process is repeated, until one of the candidates receives more than 50% of the vote after redistributions.
Of course, voters are not obliged to indicate preferences on their ballot paper, or rank all the candidates, however in marginal seats second preferences might be crucial.
The AV is used in elections to the Australian House of Representatives.
2. the Supplementary Vote system
In this variation of AV system, voter have only two preferences - they can mark a cross in the first preference column for one candidate and a cross in second preference column for a second candidate. Candidates who win more than 50% of first preferences are automatically elected.
If this doesn't happen, every candidate is eliminated, but the top two (with highest number of first preference), who remain in the race. The second preferences from the losing candidates are then redistributed. Second preferences for eliminated candidates are discarded and those for the two remaining candidates added to their total. After this, whichever of the two remaining candidates has the greatest number of votes wins the seat. Which means that the winning candidate does not necessarily need to win more than 50% of the votes cast.
This system was used during London Mayoral elections i 2000 (won by Ken Livingstone, who received 38,96% of the vote in first preference vote, but has held 57,92% of the vote in final result. )
3. the Second Ballot system
As the name implies, in the Second Ballot system, the provision of voting takes place on two separate occasions. In other words, elections often have two rounds.
In the first ballot, voters vote for their favourite candidate on the list. If any candidate wins more than 50% of the vote, then he is elected. If not, a second ballot is held, usually few weeks later.
In some variations of this system, only the top two candidates (with the highest number of votes in the first ballot) are allowed to stand for the second ballot. In other, either all candidates are allowed for the next round or only those who crossed the threshold (by winning more than 20% votes, for example).
This system is in no sense proportional. It does, however, allow qenuin choice and encourages pacts between parties.
A version of the Second Ballot system is used in France and in Polish Presidential Elections.