Sometimes the way to get to a good explanation is to start with a bad one.
The opponents of AV make the claim that it means that voters for fringe parties get their vote counted more than voters for major parties. This seemed a stupid objection, but I couldn't quite explain why, clearly and simply.
Yesterday I read John Humphrys' complete failure to explain why (via Matt Ridley), and it became obvious:
Yes, in AV, your vote can be counted more than once — whether you vote for a fringe party or a winner or runner-up. If there are only two rounds of counting in a particular example, then the person A who votes for the eliminated candidate gets their vote counted twice: for their first choice in the first round, and for their second choice in the second round.
The voter B for any other candidate also gets their vote counted twice, for their first choice both times.
So in the last round, the one that actually decides the winner, voter A gets counted for their second choice and voter B for their first.
That doesn't settle the larger argument of course: you can still argue whether AV has a tendency to produce centrist coalitions and whether that is a bad thing. But there should be no argument claiming that AV is less fair than FPTP, for what that's worth.
(Disclaimer: I argue about this out of habit, not because I think it matters)