As I mentioned a few days ago, I'm strongly in favour of electoral reform, and given the choice between the Alternative Vote (AV) and First-Past-The-Post (FPTP), I would choose the AV any day. There are plenty of good reasons for that.
But those good reasons do not include the oft-repeated mantra that the AV guarantees that MPs "would have the support of a majority". It doesn't.
First, and most obviously, there is the matter of exhausted ballot papers; I was addressing that last time.
But another issue is what we mean by "support". When I put a cross (or a number) on a ballot paper, I am not thereby expressing support for a candidate. Rather, I am expressing a preference. I might loathe all umpteen of the candidates, but I still have a responsibility to use my vote for damage-limitation (unless I spoil the ballot paper as an act of protest, or shoot all of the candidates and force a by-election with fresh nominations). So the "support" I have for my first-choice candidate might simply be that I loathe all the other candidates more than I loathe him or her. That's hardly saying much.
So, under the AV, suppose that all of the voters assign preferences to all of the candidates, so that there are no exhausted ballot papers (unrealistic, but there's no good reason not to do this). Then the MP will indeed have the "support" of a majority of the voters. But we need to be clear exactly what majority "support" means.
It means that a majority of the voters dislike at least one of the unsuccessful candidates (including the second-place candidate) more than they dislike the successful candidate. Or that they like the successful candidate more than they like at least one of the unsuccessful candidates. Or that they have a preference for the successful candidate over at least one of the unsuccessful candidates.
But not that the successful candidate has majority "support"—unless it is possible simultaneously to hate someone and be their "supporter".
So I wouldn't use the "majority support" argument, except in this sense: that the AV explicitly prevents the least-favoured candidate from getting elected. FPTP leaves this as a very real possibility, through vote splitting, which opens the door for the likes of the BNP to be elected, even when the majority would rank them last in any sensible (preferential) voting system.