Lib Dem got 3.5mil votes and have 71 seats, Sinn Fein have 210,000 and seven seats
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The UK didn’t just have one election. It had 650 elections for MP, and then the MPs decide amongst themselves who will lead.
Each of those 650 elections is winner take all, whomever gets the most votes wins.
In practice those 650 separate elections are coordinated with parties and so on.
In the US choosing the house of representatives (435 seats) or the Canadian house of Commons (338 seats currently) use the same basic system, as do others.
There are other ways to choose how to form a parliament, all of the options come with tradeoffs. Parliament itself is a majority vote system so you have to be careful that whatever electoral system you have puts people in power who can actually function as a government and doesn’t hand disproportionate power to people who are unsupported by the majority of the country, but they get outsized power by being a ‘swing’ vote within parliament or otherwise have just enough seats to prop up a majority.
One option would be purely proportional, so you would show up, vote for a party and then seats would be assigned based on the percentage of the vote. The parties then need to try and build a coalition to govern, or have more elections until they can.
Another option is to have a mix of proportional and first past the post (the current system). So you would elect a local MP. And then you would have a party vote, and then some fraction of MPs would be from a proportional list and some chosen from winner take all elections. The German system essentially tries to have a proportional outcome and so adjusts the number of seats so that the people who win local elections still sit but then the overall outcome is close to proportional.
Another option is to have ranked ballots. These can be single or multiple winners per area. Basically you rank who want to win and then whomever has the lowest vote totals is eliminated and their votes transferred to the next on their ballot. This is quite complicated to do quickly by hand, so computerised voting is strongly preferred. You do this either as a single winner system or group together multiple seats (say 3-6) and then have it work that way. The Irish have some elections like this, the biggest problem is the right number of winners per voting group can dramatically shift the outcomes and that sort of depends on things like the history of parties before you enact it.
Right now labour has 100% of the power with 34% of the vote, that’s not ideal, but if the election were proportional and people voted exactly as they had, it’s not clear who the government would be or whether or not they could maintain such a government for any length of time.
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