Can AV elect a second- or third-placed candidate?

Before the 2011 ‘Alternative Vote’ Referendum

Under First Past The Post (FPTP), everyone picks their favourite option, and the one with most votes wins. Simple, right? Under the Alternative Vote (AV), the one with fewest votes is eliminated, and this process repeats until one candidate has a majority — or until only one candidate is left; it makes no difference. No2AV will tell you that this can hand victory to the second- or third-placed candidate. But what does that mean? The winner is by definition the first-placed candidate – what they mean is that AV sometimes returns a winner who would not have won under FPTP. The Yes campaign will tell you that that is sort of the point – if AV always returned the same winner as FPTP, nobody would care which we used.

The thing is, elections are so engrained in the public consciousness that it’s hard to dislodge the idea that after one there’s a very clear “best candidate”, and any system that doesn’t elect that person is fundamentally broken. But is that true?

And what if there was more than just FPTP and AV on the ballot this May?

What if we added Runoff Voting? This is the system used in France, whereby the two most popular candidates after round one face a second round, the winner of which is duly elected. We could also add Approval Voting. This is essentially exactly the same as FPTP, but you’re allowed to vote for as many options as you like. Lastly, we could add Range Voting. With Range Voting, you can score each option, usually out of 10 or 100, and the one with the maximum score is the winner. There are still more options, but let’s stick to these five for now.

Let’s ask an imaginary population to vote on which system they want. The population is made up of:

• 40 "Reds". The reds like FPTP, but would settle for Approval. They'll have no truck with numbers or multiple rounds.
• 21 "Yellows". The yellows would like Runoff voting, and would be happy enough with AV. They would settle for Approval, but not Range or FPTP.
• 20 "Greens". The greens would like AV, and would be happy enough with Runoff. They too would settle for Approval, but not Range or FPTP.
• 19 "Blues". The blues are desperate to have Range voting, but otherwise have identical preferences to the greens.

Let’s assume nobody votes tactically.

Clearly, FPTP has most votes. So that should win, right?

Well, let’s not get ahead of ourselves. “Having the most votes” is only how you win under FPTP. If we used Runoff voting, the red and yellow vote would get FPTP and Runoff into the second round, where the blues and greens would side with Runoff, granting it a 60:40 win.

This graph shows an interesting fact: even though FPTP got the most votes in the general ballot, the population as a whole prefers Runoff. But what would happen under AV?

In round one, Approval voting would be eliminated, as nobody voted for it. In round two, Range voting, with only the 19-strong blue vote to its name, would also be eliminated. In round three, with Range gone, the blues vote like the greens, taking AV up to 39 votes. This puts Runoff into last place, and it is eliminated. Round four, therefore, is between AV and FPTP. The reds stay loyal to FPTP, but the yellows, greens and blues all prefer AV, which wins with a 60:40 majority.

Again, it is clear that the population as a whole prefers AV to FPTP. In fact, the population as a whole also prefers Approval to FPTP. This is a serious flaw with FPTP: it is possible for the “Condorcet Loser” to win. The Condorcet Loser is any candidate who would lose a two-horse race against each of the other candidates. (In our hypothetical example, however, Range Voting is the Condorcet Loser and so FPTP has not messed up as badly as it might.) AV and Runoff voting are immune from electing the Condorcet Loser, because such a candidate would by definition be defeated in the final round (if they got there).

Let’s have a look at the less well-known systems now. Range Voting is the only system to measure strength of feeling. I mentioned earlier than the blues are desperate for Range to win — so they rate it a 10. Then, they rate AV 3, Runoff 2 and Approval 1 — as do the greens. The greens have opted not to give the full 10 points to any system. That is their right. The yellows vote in a similar way, giving Runoff 3 points, AV 2 and Approval 1. The reds give 3 points to FPTP, and two to Approval. Anything I haven’t mentioned gets zero points.

In total FPTP has 120 points, all from the reds. Approval has 140 points, from everyone. Runoff slightly beats it with 141, from the yellow, green and blues. AV does a little better with 159, from the same ballot papers. But the sheer strength of feeling from the blues is enough to hand Range Voting the win, with 190 points.

Even though the majority would rather have anything but Range Voting (i.e., it is the Condorcet Loser), the theory goes that Range would make the most total happiness, albeit concentrated in a small minority of voters.

Lastly, what would happen under Approval Voting?

Only the reds approve of FPTP. Everyone else approves of AV and Runoff voting. Range gets only 19 votes, from the blues, but since everybody would be happy with Approval voting, it scores an easy win.

All five systems have elected themselves — and yet the voters’ preferences are the same in each case. Obviously this population was engineered so this would happen, and it’s very unlikely in real life. But it should illustrate some of the problems with the current system, and indeed with all systems. It is mathematically impossible to design a system without tactical voting.

Here is the issue: looking at the voter’s preferences, I don’t think there’s a single system that clearly ought to win. No2AV would tell you FPTP should win because it “came first” in terms of first-preference votes, but 60% of the population would rather have Alternative, Runoff or Approval Voting, and it’s hard to argue that democracy dictates that they mustn’t.

This makes designing a fair system almost impossible. For example, a criterion mathematicians use to judge voting systems is “independence of irrelevant alternatives”. Range and Approval Voting satisfy this criterion: if any losing system were removed from the ballot, it would not affect the scores for the remaining ones and the result would be the same. Under FPTP, removing Runoff Voting from the ballot would hand the victory to AV. In this example, AV would elect itself with any combination of opponents, but this is not generally the case.

Even though all five systems elect different winners, none of them is inherently biased. Nor do any of them give any voters more than their fair share of sway over the result, whatever No2AV might tell you. Nor, whatever Yes To Fairer Votes may have you believe, are any of them a panacea that will automatically usher in a “new politics”. We simply have to agree a system beforehand, so that going into the election everyone knows the rules. That way, while the result may not be popular, at least everybody can accept that it is fair, and knows that they have a chance to do something about it the next time around.

Can AV rob a rightful winner of victory? Yes. But so can FPTP, and and at least AV will never elect the single least popular candidate.