# maths question: how do I state the vote %ages for winning candidates in multimember constituencies?

I’m working on a project and I’ve run into an issue: how do I state the vote percentages for winning candidates in a multi-member constituency, when each voter only has one vote, in a first-three-past-the-post system?

It’s real easy, obviously, when you have a single-member constituency. You take the vote the candidate got, divide by the total votes cast, multiply by 100, and there’s your percentage.

But what happens if you’ve got a constituency that elects three members, but each voter only gets to cast one vote? You might get something like this:

Mr Able: 10,000
Ms Baker: 9,000
Mr Cook: 8,000
Ms Dumont: 5,000
Mr Edwards: 4,000
Ms Fair: 3,000
Mr Goulet: 2,500
Ms Houle: 2,000
Mr. Innis: 1,500
Ms. Jay: 1,000

If this were a single-member consitituency, it would be completely fair to say that Able’s getting elected with only 21.7% of the vote.

But it’s a three member constituency. If you just take a percentage, Mr Able looks like he’s getting elected by 21.7%, but that’s not really accurate. Because it’s a single-vote system that elects three candidates, just running the total votes seems to me to undercut the level of support he got.

Collectively, the three winners pulled in (Able, Baker and Cook) pulled in 58.7% of the votes cast. That seems a more accurate estimate of the level of their support, stating what’s needed to get elected. But I don’t think that’s really a good way to try to express the percentage, either.

Any thoughts?

You normally just state the individual vote %, and make it clear what it is that you are reporting. You can also state the turnout, and % of voters who’s candidate didn’t get in.

People who understand the voting system will understand the numbers you present, as long as you are clear about what you are reporting. People who don’t understand the voting system will also (around here) be people who reject anything other than single-member first-past-the-post voting systems, and will reject any number other than the simple individual vote anyway: you aren’t going to be able to change their mind by quoting different kinds of percentages.

No, it doesn’t. That is the level of support he got.

What’s different here is not the level of support that a candidate gets - if this were a single-seat constituency Mr Able would still be elected with only 21.7% of the vote. What’s unusual is that you don’t need to be the most popular candidate in order to be elected; it’s enough to be among the three most popular. Therefore it’s possible to get elected with a lower level of support.

It’s not possible to say in general terms what level of support is required to be elected, and therefore to express the actual vote received as a percentage of the vote required for election. The level of vote required for election a function of the number of candidates and the spread of the vote among them. But of course that is also true of a single-seat FPTP constituency. In single-seat people are generally elected with rather more than 21.7% of the vote, but that’s simply because there tend to be fewer candidates.

Understand where you are coming from and Melbourne is correct.
Use whatever is the standard metric in your jurisdiction.

Mr Able is getting elected by 21.7% of the vote.

You could calculate what % of the quota (votes cast/(elected+1) +1) they received i.e.
Mr Able: 10000 10000 87%
Ms Baker: 9,000 9000 78%
Mr Cook: 8,000 8000 70%
Ms Dumont: 5,000 5000 43%
Mr Edwards: 4,000 4000 35%
Ms Fair: 3,000 3000 26%
Mr Goulet: 2,500 2500 22%
Ms Houle: 2,000 2000 17%
Mr. Innis: 1,500 1500 13%
Ms. Jay: 1,000 1000 9%

But you are going to get candidates with more than 100% which is going to confuse all but those who understand how the count works, and they will understand the simple percentages, so there’s no advantage gained.

I should have known I’d get answers from the Aussies! You guys have experimented a lot more with voting systems. Thanks!

If they got three separate ballots, which they could cast for the same candidate if they chose to, no, the answers would not be different. The number of votes is all that matters, not the number of voters casting them.

What if the voter gets one ballot, but can put an ‘x’ beside three names? Obviously the maximum that any candidate could achieve would be 33.3% of the vote, and there might be some value in expressing each candidate’s actual vote a in relation to that theoretical maximum attainable personal vote.

They could get a maximum of 33.3% of the possible votes, but more of the votes actually cast. People who feel strongly for one particular candidate have an incentive to not use all of their votes. This isn’t a great example since there were no losers, but a council race in my city a few years ago had three candidates who got 35%, 34%, and 31% of the votes cast.

In theory, yes. In practice, party politics militates against this.

The “each voter has as many votes as there are seats to be filled” model is a pretty stupid. If the Sensible Party expects or hopes to secure the support of >50% of voters, then the rational strategy is for the Sensible Party to run as many candidates as there are seats to be filled. If Sensible Party voters are loyal, anything above 50% of the votes can win 100% of the seats for the Sensible Party, regardless of the number of seats to be filled.

Thus the only reason for an individual voter not to use all his votes is a desire to advantage a specific Sensible Party candidate, even at the cost of disadvantaging other candidates from the Sensible Party, with the result of a seat being taken by a Silly Party candidate. This strategy only attracts voters who have no party preference whatsoever.

How may seats were to be filled? Was it by any chance three? And were the three candidates you mention all by any chance from the same party or allied parties?

Four, actually. And it was a nonpartisan election (actually nonpartisan, not just with the party names hidden as happens in some elections). Like I said, it’s a bad example, but I know of other past races where a candidate specifically advocated a just-for-me vote.

I must admit I was unaware that this system was actually in use anywhere. It seems to have no merits of any kind; why would anyone adopt it?

I note that some of the candidates in the election you mention advocated a “just for me” vote. As a matter of interest, have we any measurement of how successful that advocacy was. We would know this if we could compare the number of voters who voted with the number of votes cast.

Thanks for the feedback everyone.

I’ve now run the actual numbers. Nineteen candidates for four seats, with five different parties, plus two independents. The top four candidates each got in the range of 11% of the votes, give it take some decimal points. Looks odd but it’s done.

Check!

Ditto!
A third method would be if they put 1,2 & 3 beside their selected three candidates and then you are away with the preference voting band wagon.
i.e. should the candidate who got say 5,000 votes all ranked as #1 be elected over the candidate who got say 6,000 votes all ranked as #3.
Again various voting systems can give either result. So pick the preferred electoral system, pay your money and take your chances.
If there is any advantage in tactical voting by the second election the candidates will be all over it like seagulls after chips.

If this were a single-member constituency in Australia, with a single transferrable vote, then any of the candidates could be elected, except for Ms Jay, since she would be eliminated at the second count.

If this were a multi-member constituency, as in the Australian Senate, the lower house in the Tasmanian Parliament, and other Australian elected bodies, then again any of the candidates could be elected, except again for Ms Jay (since no candidate got a quota of 11,501 on the first count), and none of the candidates would be sure to get elected. (If one or more of the candidates got more than a quota on the first count, then they would be certain to be elected, and even Ms Jay would have a chance of being elected, if she received enough preferences from leading candidate(s).)

So what is meaningful as a percentage depends on the system used to count votes. With the proportional representation systems used for multi-member constituencies in Australia, the most meaningful number is the percentage of the quota received by a candidate or party.

I don’t understand why it looks wrong to list either the number of votes or the percentage each candidate got. As long as it’s clearly stated that the top three get elected, if necessary presenting the list with a clear line separating the top three from the rest, what’s the problem?

If you’re trying to indicate a legitimacy that would somehow equate to getting more than 50% of the vote, that just isn’t there on this distribution of votes, and fiddling about with the figures won’t make it so. Whether that then suggests there’s some legitimacy lacking in the fact that people can be elected with such a small share of the vote (as indeed would be the case if one candidate got 50+% and the next winning two might have a correspondingly smaller share) is a separate value judgement.

True
But either Houle or Innis will be eliminated in the third round because Jay doesn’t have enough votes to allocate and get them both past Goulet.

Then it gets progressively more complicated.

The numbers are accurate. What you haven’t got is a measure of the unfairness .

To what extent do similar minded candidates split the vote … How many would candidate A get if candidate B had not run … you can’t know because you didnt get the voters 2nd choice.
England uses first past the post and the talk is how two candidates split the vote and cause them to both lose…
Australia uses proportional preferential voting. Voters number their candidates in order . When candidate C is lowest and therefore knocked out, then votes for him are redistributed proportionally according to the next vote on each slip that is for a candidates that is still in the running. So if he recieves 10% of the 1’s, his pile counts for 10%. Then if 75% of votes for him go to candidate A NEXT (eg 2nd , but basically the next lowest numbered still in the running candidate ) , then 10% of 75%, ie 7.5% go to candidate A.