The main argument against this system is that in an election with more than two candidates, it may result in the election of a candidate who has received only a minority of the votes cast, so most of the people the MP is representing didn't vote for him/her.
Parties which come second or third often get large numbers of votes but few seats, so smaller parties don't get many MPs in the House of Commons. For example in 1992 the Liberal Democrats won 18% of the votes and 3% of the seats.
In the same way it is possible for a party to win most seats with a minority of votes, but fewer than one of its rivals.
Some seats are so 'safe' for one party that supporters of another group have a vote which won't make any difference to the outcome. It is therefore wasted.
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