The 'Condorcet criterion' is a reason why AV is better than FPTP.
The Condorcet Winner of an election is the candidate who, if they took part in a two-horse race against each other candidate, would win each of those elections. It should be obvious why this is a desirable quality- it demonstrates that that candidate is the 'best' for a sensible definition of the term.
Neither AV nor FPTP are guaranteed to select the Condorcet Winner, but AV is more likely to.
There's also a counterpart- the Condorcet Loser criterion. The Condorcet Loser is the candidate who, in a series of elections against the other candidate to win. That is, a universally unliked candidate. The Condorcet Loser can win a FPTP election, but can't win an AV election.
AV is far from perfect- there's actually an anomaly that can occur occasionally where downrating your preference for a candidate can actually help them, but a little analysis has shown that maybe 7 out of the 600-odd seats in the UK might suffer from it. What is unambiguous, is that FPTP creates far more troubling anomalies far more frequently.
Yes to AV, because it is a mathematically superior system.
Opportunities to change how democracy is conducted don't come around very often.
Perhaps the most significant reason is because they typically require a large level of support from the ruling party. It's only going to support a change if the change benefits itself- if it helps keep it in power.
More representative systems typically leech power away from the currently dominant party- FPTP is not a very representative system, so any proposed move away from it would hurt the current party in power. It also has the mathematical property of exaggerating the difference in support for parties, so even a modest plurality of votes overall can lead to a massive majority of seats. So it's very rare that a government would countenance it because even a large majority might disappear at the next election under a better system. So this is a rare opportunity to improve how elections are conducted in the UK.
If AV doesn't get adopted, there may never be another chance to move to a different system in our lifetimes.
If it does get adopted, we may have to wait a while before further reform, but the case for it will be much stronger. If you prefer a different system to AV, your best chance of getting it at all is to vote for AV now.
Yes to AV because improvement has to start somewhere.
The No2AV campaign has been dishonest on a scale even beyond what I've seen in pamphlets for the BNP and UKIP. I fervently believe that no-one involved in the creation of the campaign should be permitted any political appointment in the future. In my ideal world, the electoral commission would bar them from standing for anything. It makes me physically ill.
It will produce more coalitions.
Objectively false. Studying past elections in the UK shows that it would have led to hung parliaments no more frequently.
It is used by only three other countries
Objectively false. Nine, actually, including the USA and the UK for some elected positions (such as in trade unions). The fact that it's under-represented in parliamentary elections is not an argument against it.
It allows the second or third placed candidate to win
Under AV, the person who won under the rules of AV wins. Under FPTP, the person who run under the rules of FPTP wins. Sometimes these two are not the same. There is nothing inherently more accurate about FPTP. In fact, according to things like the Condorcet criterion, FPTP is objectively worse.
It will cost the country £250 million
Objectively false. Of the quoted £247m, not a single penny is actually true. The referendum itself is £91m, a sunk cost. Voting machines might cost £130m, but AV can still be hand counted and machines might be used under FPTP eventually anyway. Explaining AV to voters will not cost £26m. You can explain what to do in a single sentence. Further information is available in many places on the web, for free.
Remember the core principle of British democracy: every person gets an equal vote.
Except FPTP guarantees that not everybody's vote is equal. It's actually possible to calculate how much an average vote is worth in a particular constituency based on the size of the standing majority- it's never 1. Under AV, the 'value' of your vote improves. There is nothing in AV - nothing - that changes One Person One Vote. AV is called 'Instant Run-Off' in the US, because that's what it is. It's a much more efficient way of having a series of elimination rounds to get a winning candidate. If your candidate is eliminated, it is counted differently in subsequent rounds. If not, your vote is counted for the same person in future. At no point is any person's vote tallied more than any other person's.
I'm used to lies in political material. It's par for the course. But usually it's something that's at least plausible, an opinion or a prediction. It's usually difficult if not impossible to objectively prove it's false. Every claim made by the No2AV campaign is objectively false, or deliberately misleading. If No2AV succeeds, this is what we can expect out of electioneering for the future, and it stinks.
Yes to AV, because the people behind the No campaign deserve to be publicly humiliated.