Firstly, before the system can be implemented, the constituencies we use now have to quite radically changed. In a kind of super-border change, every constituency would be merged with another (possible three into one). The most obvious observation is that you would have a third of the MP's we had before, and indeed there would a 2/3 reduction in winners for each constituency. However, where STV differs from the current system, is that each constituency has three winners.
In the current system, whoever wins the seat, has by definition 100% of the seats for that area. This means that even if the people voted and lost by only a single % for their chosen candidate, their opinions would not be expressed whatsoever in parliament. Votes not for the winner, in our system, are effectively wasted. In comes STV, where everybody has a greater contribution on the whole to the winner(s).
If we wanted multiple winners, there are different ways you could have it. You could for example have the top three % votes be the ones who receive seats. But, the thinking goes, is this actually how the people would want it? Let's say for example you want Labour to win, but you would be happy with the liberals. You vote labour. Others echo your feelings on the other side, voting conservative but being happy with the liberals. But because neither of you actually voted liberals, they receive no votes. The system is essentially a measure of die hard conservatives and labourites. Who would waste their vote on a second choice? The other side may get in! In the end the only people voting liberal are the ones not scared of the either of the two getting in. While liberal may place third and get their seat, the system leaves this completely in the air, even though the party would have the greatest overall approval of the population (they are not hated by the people voting any other party for). Furthermore, there would just be a constant race for third place between all labour/conservative candidates, even if the vast majority of people hated those parties.
STV is an attempt to make that system much fairer. As you may have guessed from the description, the implementation of a "I'd be happy with X" vote is the key. The "transferable" in the name is also a nice indicator to the more literal readers.
In the transferable vote system, your ballot paper will have a two vote priority system. To repeat the vote from before, people will now be able to pick liberal democrats as their second choice. To decide the first seat, whoever gains the required seats wins. If your party wins, hooray. But it is likely they won by more then just one vote. Every vote the winner has on top of the required is now moved to the voter's second choice. If we had a hundred people who votes labour and liberal, and labour only needed 50 votes to win, then the other fifty are now given to the liberals. The new totals are then compared (labour doesn't take part since they already have a seat), and the person with the required votes gets the second seat. As before, the votes for the winner past the win point are moved to the people's second choice.
Obviously its repeated again, for the last seat, and you can see how a wider part of people's opinions are being used. Sufficed to say (trust me), if nobody wins a round, a slightly more confusing bit of maths happens, but the results will be the same; the three parties most wanted overall are the ones who receive seats.
Past the constituent level, the system is the same; parliament is still made of MP's and parties, voting together against each other, the Prime Minister is the party leader with the most party seats, and there's not a noticeable change in those mechanics.
However, there is one important change, totally relevant to the current situation. If we are going to multi-party vote, and parties like the liberal democrats are going to gain greater seats representative of support, we are going to hang parliament a lot. And I mean near constantly. The reason we almost always have a majority, is that with two parties massively bigger then the rest, its likely one of them is going to pass the magic 50% mark. Half votes to each party are 50% after all, and just a single seat wins it for one of them. With any form of greater voter representation, you are going to get more than two parties. And once you have three parties, equal voting gets you 33%, which is very far from a majority.
And since legislation requires a majority, parties would have to team up to pass anything (any two of those parties are way above the majority), and so cooperation and deals are very common. Some say proportional representation systems create stagnant governments, since change becomes rather difficult, and the changes that do take place involve back handed deals between parties.
God forbid there be discussion in parliament. Personally, a system that constantly yields majorities seems rather silly to me. Since you can pass whatever you want, the entirety of the other members of parliament are accomplishing little with their time and pay. In the last government, for example, conservatives could do nothing to stop labour passing a law, and their only input was constantly saying everything was a bad idea. If the conservatives had won this election, they would have been able to do anything they wanted until the next election, in which case it may switch. The function of the opposition is just to gain support from the dissenters, and represent that in debates (debates with no actual weight to them). And since we've all seen how fickle the people are, does that temporary dissent really matter much?
I'm of the opinion the current system is of greater stagnation, it just comes in seasons of red and blue where they take turns undoing each other. We've seen in this election the great support for liberal democrats wane, seemingly because everybody chickened out voting for someone other then the two parties, since the one they hate may have a chance of getting in. Indeed, the system we have is such that liberal democrats gained more votes this election, but lost a significant number of seats.
To show you what I mean, and provide a close, these are the vote-to-seats this election saw. Taken from the bbc coverage as of this moment, where there are 649 seats in this parliament.
Conservative, 36.1% of the vote, 306 of the seats
Labour, 29% of the vote, 258 of the seats
Liberal Democrats 23% of the vote, 57 Seats
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