I don't often do a post which essentially just links to another blog post but I am making an exception here.
This is simply one of the best blog posts I have ever read from award winning mathematician Tim Gowers. He goes through all of the arguments for and against AV and FPTP and with intelligence and humour analyses each one from a mathematical perspective but in an accessible way. He is clearly in favour of a switch to AV but he is not afraid to point out its weaknesses as well as its strengths.
It is a very long post and will probably take 20 or 30 minutes to read in full but it is very much worth it. If you are still undecided about AV (and even if you think you aren't) I heartily recommend it.