Thursday, 7 April 2011

Alternative Vote: How big a deal is non-monotonicity?

One of the few areas in which Alternative Vote is unequivocally less good than First Past The Post is "monotonicity". Here's the recap explaining what "monotonicity" is, why it's generally a good thing for voting systems to do, and why AV doesn't guarantee it.

I said in that post that it wasn't generally a problem in practice, or no-one would actually use AV. Of course, one person's definition of "not a problem" might be "no more than 1 in 20 elections", while another's might be "no more than 1 in 1000 elections". Depending on how large the effect is, it might be unimportant compared with the disadvantages of FPTP over AV, or it might overwhelm them.

So, I've modified the AV swingometer to, in addition to reporting which seats change hands under a particular set of votes, to also report on whether any of the seats had a potentially non-monotonic set of preferences. Experiment for yourself to see the real scale of the problem.

More details about the methodology and the results below. In summary: it's very rare - neither non-monotonicity nor the tactical voting opportunities it creates will be a practical problem for AV elections.

Methodology for detecting non-monotonicity

There are two types of monotonicity-breaking election.

The first type is fairly easy to define, and is the one given in the example in the previous post. A winning candidate gains extra votes from a rival (or rivals), which changes the order in which candidates are eliminated, which causes them to lose the election despite their standing on every individual ballot paper being the same as before or better.

To detect this, in a situation where the original order is "A, B, C", the swingometer changes just enough B first preference votes into A first preference votes to change the order to "A, C, B". If A then loses, the election is non-monotonic.

The second type is where this has "already happened", and a losing candidate could, by losing votes to other candidates, adjust the elimination order in their favour, and win the election. (They could, of course, also win in this situation by gaining more votes and win by having sufficient first preferences to scrape 50% on the current transfer order)

To detect this, in a situation where the original AV order is "A, B, C", but with B ahead of A on first preferences, the swingometer changes just enough B first preference votes into C first preference votes to change the first preference order to "B, C, A". If B then wins instead of A, the election is non-monotonic. (If giving these extra votes from B to C causes C to win rather than A, the election is still monotonic)

There are other ways in which transfers could cause non-monotonicity, but an election with at most 3 viable candidates is too simple to bring them up.

I only change transfer values in the detection step - with only three significant candidates, improving the transfer rates in favour of the winning candidate can only reduce the size of the monotonicity window.1

I then also measure the size of the monotonicity window by considering how many more votes the affected candidate would have to gain or lose to change the result back to the expected one - the windows are fairly narrow, so the situation is usually "no change" is a win, "small change" is a loss, "larger change" is a win again.


Of the 531 seats considered by the swingometer, a potentially non-monotonic result occurs is very few. Exactly how many depends on the first preferences and transfer weightings, of course, but it's generally low. There are only 6 on the previous general election results, only one (a different one) on the 6 July 2011 polling, and none at all on the June 2010 polls.

The swingometer will tell you the estimated first preferences for the seat, the smallest change to those first preferences to give a non-monotonic result, the party that benefits, and the size of the non-monotonicity window for each seat. The windows are often less than a thousand votes, and very rarely more than three thousand votes.

(These are worst-case sizes, too - a candidate whose second preferences also improve or worsen as their first preferences do will have an even narrower or perhaps non-existent window)


In the vast majority of seats, non-monotonicity will not be a problem at all. In a few seats (generally a specific sort of three-way marginal) it does occur. However, the size of the non-monotonicity window is relatively small even then - it's more likely that a candidate will go straight over it than land within it.

For tactical voting under AV, the situation is even better. Tactical voting under AV is basically about exploiting a non-monotonic situation to your advantage. To do this, you need to do two things:

  • Identify the direction of tactical vote you need to encourage
  • Calculate the size of the non-monotonicity window, and vote accordingly.

The problem is that whether the window exists at all, and how big it is, depends very much on the exact totals within the constituency. National opinion polls have a 3% margin of error for first preferences2, and much more for second preferences. Applying national poll results to individual constituency is even less reliable - Uniform National Swing (and the Uniform Proportional Swing variant used in the swingometer) are reasonably accurate for predicting overall results, because errors in individual constituencies tend to cancel out, but they're fairly bad at predicting the result of individual constituencies, and extremely bad at predicting the vote totals within those constituencies.

A relatively small change in first preference votes and transfers - around 1% -

Polls of individual constituencies are extremely rare, expensive, less accurate than national polling, and would need to be carried out very close to polling day to be any use.

Because the tactical voting relies on doing something unfavourable to your party, attempting and failing could mean that rather than hitting the (non-existent) non-monotonicity window, you instead throw away a winning position by weakening your preferences. It's therefore an extremely high risk strategy in most circumstances.

Neither non-monotonicity nor tactical voting is therefore likely to be a problem in practice for English use of AV. I suspect the same will also be true in the other three nations of the UK, but I don't have the data to prove that.


1 Moving transfer rates against the winning candidate of course makes it more likely that they'll lose, but then it no longer proves non-monotonicity if they do, because not every vote has either stayed the same or changed in their favour.

2 In addition to this, there is an unknown systematic error, though with a reputable polling company this should be fairly small.