The alternative vote, which is not a system of proportional representation (it would still mean no Green or BNP MPs), usually has one advantage over first past the post: people need not vote tactically. They can vote for their first preference, and if that candidate is eliminated, transfer their vote to another.
That usually works, but not always. In constituency X the first past the post results is as follows.
The centre candidate has only won perhaps because 10% of left voters have voted tactically for the centre. Without tactical voting the first round would be.
Under AV the centre candidate is now eliminated and if his vote transfers equally between left and right the right candidate wins.
We can thus see that AV neither leads to proportional representation, nor does it necessarily lead to MPs more closely reflecting voters’ wishes. It’s a marginal improvement, if that.