Take a look at the blog post here by "dbirkin". It's worth a look as I think that in all honesty it is probably the most confusing aspect of the AV vs FPTP debate. It comes down to things I've spoken about before, the lack of a Condorcet method, for ensuring the winner is always the most popular, on the table. It's easy to use it against AV, but to do so without realising that it's also an argument against FPTP is false.
The problem with DBirkin's argument is he's only looking at one half of the problem, that of AV's ability to hit a specific scenario whereby the "most supported" is actually the third place candidate overall. The whole picture needs to be told to be fair to those that are trying to understand the advantages and disadvantages of each system.
This scenario mentioned above requires, statistically, for more people from the first and/or second placed parties to support the third party as their second preferences for a combined total, than the combined number of their rival's first preferences with the transfer of the third party's second preferences.
To try and visualise this, see the preference table below.
20 A voters, second preferences - 20 for C.
15 B voters, second preferences - 15 for B.
10 C voters, second preferences - 1 for A, 9 for B
In this scenario a potential vote score of 30 is available for C, and this can't be beaten by a combination of B's voters and C's 9 preferences moving to B.
However change it up a bit, to be a little bit more realistic.
20 A voters, second preferences - 10 for C, 10 don't transfer.
15 B voters, second preferences - 8 for B, 7 don't transfer.
10 C voters, second preferences - 4 for A, 4 for B, 2 don't transfer.
In this scenario the potential score for C is only 20, whereas A will have 24 by comparison, being a worthy winner.
The assertion by DBirkin here is that the scenario at the top means the whole system is worse than FPTP. So. How would it work under FPTP?
20 A voters, 15 B voters, and 10 C voters. Unfortunately, as DBirkin admits, FPTP is ignorant. We don't know specifically who people are voting for and why. But let's take one possible example, using the same figures as a FPTP scenario.
of the 20 voters for A 10 are true supporters, 2 are ex -B voters, 8 are C voters...but all vote (tactically or not) to give A support.
Of the 15 B voters, 14 want B and 1 would rather have C but don't like the rise of A support and want to try and stop it.
Of the 10 C voters all are primarily C voters.
This gives a "real" first preference count (for example, under AV) of...
12 A supporters
14 B supporters
19 C supporters.
Yet the system is such that enough people have to guess about how people are going to vote that they actually turn their first place support in to last place ranking.
The reality is that far from saying "FPTP is ignorant, but AV is a lie"...actually FPTP is ignorant, and it's ALSO a lie (we can never tell under FPTP when it's lying to us or not outside of safe seats, unlike AV!). AV can certainly be a lie...but at least it's a white lie.
What do I mean by this, white lie?
FPTP can result not only in the most preferred candidate losing, but in the LEAST preferred candidate winning (in terms of first preferences). This is the travesty of the broken system of FPTP, delivering results that are actually the opposite of what people initially want.
With AV we see more honest initial preferences, though this can still lead to DBirkin's "lie" scenario. However unlike FPTP there is no chance of the initially most hated party being voted in to power, they are eliminated in the first round. We may end up with an MP that is not the most popular, but that happens every election under FPTP currently. It's not ideal, but at least we know that the winner is one of the most popular to begin with!
So under AV, the lie can be no worse than what we have currently, and in certain situations could be more representative, or endorsed, than the lie we have to live with under FPTP. In even more situations AV will deliver the candidate who is, on balance, the most endorsed, and isn't a lie at all.
To use this argument of results being a "lie" against AV is to also use it against FPTP. It is not an argument against reform, but one of criticising the practice of using non-Condorcet methods...of which both AV and FPTP can be described. We don't have a Condorcet method on the table, we have only AV or FPTP; In that comparison it's clear which system provides the more fair results.
AV a lie? No more so than FPTP already is, and no where near as ignorant or potentially unrepresentative on a local level.