The latest line of attack from the No campaign is that BNP second preferences could end up deciding elections, and this would cause the BNP and other extremists, while having no chance of winning the seat directly, to have influence on the election because their voters' second preferences would need to be attracted.
The implication is that therefore politicans would pander to BNP voters more than they already do.
This is not the case. The circumstances in which BNP second preferences could decide an election are very rare - and crucially, equivalent circumstances could occur under First Past The Post
For BNP votes to be decisive, you would need a constituency where all of the following conditions were true:
- On the final round of counting, the winner was over 50% of remaining votes by a margin significantly smaller than the number of BNP first preference votes.
- All votes except for the winner and runner-up have been transferred (otherwise, another party's transfers would overwhelm any effect from the BNP transfers)
- The majority of BNP first-preference voters expressed a preference for at least one of the remaining two candidates, and the winner of the election benefited from those preferences significantly more than the other candidates.
In that rare case, it's possible that transfers from the BNP could make the difference between being first and being second.
However, that's also true under First Past The Post.
"What?" you say - "first past the post doesn't have transfers" - well, it does, in a way. All first preference votes for parties not standing in the election are transferred before voting starts. (In my constituency in the last general election, no party I actually liked was on the ballot paper, so I was on to second or maybe even third preferences before I even got to the polling booth)
Consider a constituency like Bolton West where the BNP didn't stand a candidate. The margin of victory there is 0.2% - and in nearby constituencies where they did stand, the BNP are getting around 5% of the vote.
If BNP second preferences favoured Labour over Conservatives, then Bolton West was almost certainly only won by Labour because of BNP second preferences in 2010.
(Conversely, if BNP second preferences favoured Conservatives over Labour, then Hendon was only won by the Conservatives because of BNP second preferences)
Another obvious case - Oxford West and Abingdon where if BNP second preferences in 2010 favoured Conservative over Lib Dem they almost certainly changed the result in the seat.
In fact, there were 13 constituencies (excluding Northern Ireland) where the BNP didn't stand a candidate, but the margin of victory for the winning candidate was below the 2% the BNP candidate would probably have achieved. The second preferences - due to lack of candidates - of BNP voters almost certainly changed the result in at least some of those seats compared with a situation where the BNP had stood.
So that's the case where the BNP aren't standing. What about the case where they are?
In Solihull, the Lib Dems beat the Conservatives by 0.3%, in an election where the BNP had 2.9% of the vote. Had the Conservatives convinced a few more BNP voters to vote for them, then the BNP would have had only 2.5% of the vote, and the Conservatives would have retained the seat by 0.1%.
In a close FPTP election, parties need to convince "natural" voters for third and fourth parties in the seat to vote for them instead - the Lib Dems are (in)famous for their bar charts. In a close AV election, parties need to convince exactly the same voters to give them a better later preference than their rival gets.
There were 16 seats at the last general election where the margin of victory was smaller than the number of votes for the BNP. In 6 of them, fewer than a quarter of BNP voters would need to have been persuaded to vote for the runner-up instead to change the result.
In both voting systems the need to reach out to voters who are not your "natural" voter is there. Indeed, this is true of every voting system, since the point of elections is to convince people to vote for you more than they vote for the other candidates.
Politicians will always pander to pretty much everyone they think they can pander to without losing more votes than they gain.
The BNP - while electorally unsuccessful - have been quite successful under FPTP in making racism and especially anti-immigration rhetoric part of mainstream political dialogue: shifting the debate while apparently losing it.
It's possible under AV, where they stand no chance of winning a seat - which is why the BNP is one of the few political parties officially opposing AV - that their influence would be reduced. That's hard to predict, and I doubt it would happen to any noticeable extent - but it's certainly true that their influence wouldn't increase under AV.