Fairness in voting

In less than a week, Australians will vote for their choice of prime minister in a federal election. I have always been interested in fairness but where voting is concerned, it can be very difficult to devise a fair voting system. Australia uses a preferential ballot system where voters rank candidates in order of preference. I’m not exactly sure of the finer details of the Australian system, but I want to illustrate problems that can arise with voting systems where candidates are ranked in order of preference.

Consider the following system (this is not the same as Australia’s preferential ballot system). Let’s say there are three candidates, Bob, Ann and Mary. For simplicity, this example will have just five voters. Voters must rank Bob, Ann and Mary in order of their preference. Here’s how our voters (A,B,C,D and E) voted:

Bob Ann Mary
A 1 2 3
B 1 2 3
C 1 2 3
D 3 2 1
E 3 2 1

If we assign points to each preference, so a candidate placed first gets 2 points, a candidate placed second gets 1 point and a candidate placed third gets 0 points, we find that Bob wins the election with the tally as follows:

Bob – 6
Ann – 5
Mary – 4

This all looks fairly reasonable as Bob got the most votes for first place so it stands to reason that he should come first. The only odd thing is that Ann, who was not placed first by any of the voters comes in second, while Mary, who was placed first by 2/5 voters, comes last.

But what happens if we had a fourth candidate, Jo, and we don’t change the order in which people voted for Bob, Ann and Mary but instead slot Jo in so that the order of the other three remains the same?

Bob Ann Mary Jo
A 1 2 3 4
B 1 2 3 4
C 1 2 3 4
D 4 2 1 3
E 4 2 1 3

Now the points go like this:
First place – 3 points
Second place – 2 points
Third place – 1 point
Fourth place – 0 points

The tally is as follows:
Bob – 9 points
Ann – 10 points
Mary – 9 points
Jo – 2 points

Suddenly Ann is in first place even though no-one put her as their first preference whereas Bob, who has the most first votes, now ties second with Mary.

What does this mean for people voting in the upcoming Australian Federal election? It should go without saying but my recommendation is to be thoughtful about all of your preferences and if there’s someone you really don’t like, put them last.

17 thoughts on “Fairness in voting”

  1. Very interesting example Rachel but that is not how our preferential system works. Each vote is worth just one vote, if no candidate gets more than half the votes, then the last candidate’s preferences get reallocated according to their stated preference. If this does not give the leading candidate more than half, then the next last candidate’s votes get reallocated, and so on, until someone ends up with a majority. In Australia, there are two main parties, so often it is a contest between the candidates for those parties with the minor parties losing out, but not always. There is always a lot of behind the scenes activity to agree where preferences will go. These preferences are stitched by the party hacks and the voters get little say, unless they decide to vote independently of the party ticket.

      1. Good comment by Max. Interestingly, in Queensland, the Peter Beattie Labor government introduced optional preferential voting about ten years ago for the State election hoping to reduce the exchange of preferences between the two conservative parties, the Liberals and Nationals. It worked for a time in his favour until these two conservative parties merged.
        There are arguments for both first-past-the-post and preferential voting. All things considered, I believe preferential is the fairer system as the successful candidate will always hold the majority of votes even when his primary vote was in a minority, say 35% of the total vote.
        If you really want to complicate matters, have a look at voting for Senate seats. Quotas, and then preferences, come into play.

  2. As Kenneth Arrow’s Impossibility Theorem establishes, the Australian preferential voting system (of which there are several minor variants) can, like the one that Rachel described, also lead to bizarre results.

    There are concerns in this election that Arthur Sinodinos, the highly respected and third-ranking Liberal candidate in New South Wales, may lose to xenophobic maverick Pauline Hanson. This would happen if Hanson gets more than 2.25% of NSW senate first preference votes – which she could well do. Six NSW senate seats are being contested this time and to be successful a candidate requires 14.3% of the votes after distribution of preferences. An observer from Mars would find it astonishing that a complicated cross-trade in preferences can boost 2.25% of the vote to 14.3%, as do I, http://www.smh.com.au/federal-politics/federal-election-2013/hanson-poses-a-real-threat-to-sinodinos-cabinet-hopes-20130828-2squo.html.

    I don’ think this result would really reflect the will of the electorate. If there was a straight-out first-past-the-post poll for Sinodinos versus Hanson, Sinodinos would win in a landslide.

    Arrow is 92 this year and may not be following this election. If he is, he may be pleased to see the unintuitive result of his theorem displayed in this interesting practical example.

    1. Thanks for the input, MikeM. I thought you might have something to say about this post and was hoping you would comment.

      If anyone is interested in Arrow’s impossibility theorem, here’s a cut and paste from wikipedia:

      In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives (options), no rank order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specific set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.


    2. But, Mike, it’s still a majority of sorts and a chance for minor parties to have a say in government….none of The Green senators, for example, would secure a Senate seat without preferences. Would they? In the end it seems to make it almost impossible for the government of the day to control the Senate.
      Currently in South Australia, the speculation is seemingly endless – who will secure the sixth seat – Independent, Nick Xenophon, or The Greens Sarah Hanson-Young (aka SHY)?
      Surely you remember the preferential system from your own days as a voter in Oz, Rachel? Experience is the best teacher.
      Perhaps, the optional preferential system introduced by Beattie to Queensland would better reflect the wishes of an electorate and not lead to such bizarre results?

  3. I’m not arguing that the Australian preferential voting system be dropped. (As Arrow showed, every possible voting system for a field of more than two candidates, including optional preferential voting, will occasionally throw up perverse results. However different systems are perverse in different ways.)

    If Hanson achieves a senate seat in the way discussed above, it’s not a majority of any sort at all. Firstly, 14.7% of the vote is not a majority. Secondly, due to the opaque way in which preference deals work, most of the votes that Hanson ends up with will belong to people who had no idea that that is who they ended up voting for.

    This would not be the first time that a senate seat was won by someone with negligible public support. In the 2004 federal election Family First Party’s Steve Fielding obtained a seat in Victoria with 1.9% of the first preference vote, http://en.wikipedia.org/wiki/Steve_Fielding

    1. I agree. It is the best system. It’s a pity so many people don’t appreciate how lucky we are to live in a democracy. The number of eligible young voters (18-24 years) who failed to enrol for this election is estimated to be about 400,000 according to the AEC.

      1. Much of it is essentially saying what Mike M has already said above.
        From Boswell, “These minor-party preference deals could give Senate seats to individuals receiving little more than 2 per cent of first-preference votes….”. He also says he agrees with the respected election analyst, Antony Green, that “the Senate election is an outrageous fiddling of the electoral system” and “an international laughing-stock” that must be changed.”.
        And as you were suggesting yourself, Rachel, if a voter isn’t careful they may end up electing someone with completely different policies to the person they cast their vote for.
        To re-iterate some of Boswell’s article, “-people should cast their vote for one of the major parties, one of the parties that can actually govern.
        The alternative invites anarchy and chaos.”

  4. Sorry to add this but you were initially talking about the preferential system for The House of Reps, Rachel, and not the Senate. In the case of the latter, even if one is very thoughtful about whom one votes for, as you were suggesting, the cross-reference deals among minor parties may mean the voter has no idea where his vote will eventually end up electing!
    Your post has made me aware the system, in The Senate, needs an overhaul. Thank you.

    1. I’m not sure that my example applies to the Australian system which is quite different, although still a preferential system. I just wanted to highlight that sometimes weird results can come about from a seemingly straight-forward approach to counting votes, something I find really interesting. These are problems in mathematics that mathematicians are currently working on.

      1. Well, your blog has caused me to look more closely at our system and I must admit that I am now in agreement with Antony Green. So thanks for that.

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