In Australian elections, only 5 per cent go below the line in upper-house ballots, using precious Saturday minutes to make sure the candidates are in order of best to worst. The rest stay above the line, choosing to leave it to their first choice of group ticket to determine where their vote goes.

Sometimes these votes end up in unexpected places, such as when the Motoring Enthusiasts Party took a seat in last year's federal election, despite getting only 0.5 per cent of first-preference votes. That this took everyone by surprise suggests there is something chaotic and interesting going on in our electoral system.

As happens in many countries, the Australian upper-house system works by knocking out the lowest-scoring candidate, distributing these votes to candidates still in the running, and then repeating this until all seats are filled. This procedure is lengthy and can entail recounts, which often take days to finalise.

In the federal election, the Motoring Enthusiasts Party scraped into a seat after no fewer than 37 transfers. A vital factor in this win was that several minor parties agreed to preference each other ahead of the major parties. This can lead to strange outcomes and the question remains as to how common they are.

Intuitively, it can be argued that unexpected results are more common with above-the-line voting because of the large number of votes being shuffled around when a party is knocked out. It is worth using a mathematical model to find out if this is correct.

To test the Australian voting system, we modelled a scenario with 10 parties – labelled A to J – competing for six seats. The first preferences are shown as percentages in Figure 1.

Since the quota is 14.3 per cent, major parties A and B – with 25 per cent of the vote each – win a seat automatically. The remaining four seats depend on preference flow. We assume that each below-the-line voter places the remaining parties in a random order.

Similarly, for above-the-line voters, each party will choose flow-on preferences randomly. Although this ignores ideology and strategy, which are too complex to model, randomisation gives us a level playing field to compare the two kinds of voting.

The average number of seats won by each party is shown in Figure 1 – in the case of 1000 voters over 10,000 trials, comparing an all-above-the-line election to all-below. With all below-the-line voting, in 66 per cent of cases parties A and B win two seats each and parties C and D win one. In 34 per cent of cases, the minor party E nabs one of these seats owing to favourable alignments in voter preferences.

With all above-the-line voting, the minor parties E to G all have a substantially increased chance of winning a seat. Parties A to D win all the seats only 18 per cent of the time. The average number of seats won by each party is closer to reflecting the first-preference percentages in the above-the-line case. Above-the-line voting, where no second preference is given, better captures the original first preferences than below-the-line voting, where everything is written out in detail.

To further investigate how above or below-the-line voting affects outcome, we studied what happened when a percentage R voted above the line while the others voted below. For each value of R, we repeated the election 10,000 times to find the average number of seats won by each party.

In Figure 2, we note how the minor parties' chances depend on R. For large enough above-the-line voting, even parties H and I – with 2 per cent of first-preference votes each – can get a foot in the door, although winning a seat is still quite unlikely.

So far, we have shown that with above-the-line voting, minor parties are more likely to end up with seats. Perhaps a more important property is stability – that is, whether a small change to the vote makes a significant change to the result. In our case, a simple way to simulate a small change is to imagine that party J drops out of the election.

Although this party would always be knocked out in the first round, this changes the fate of up to 1 per cent of votes. Shown below is the likelihood that this changes the results of the election. This 1 per cent of votes clearly becomes crucial to determining the outcome of the election when there is significant above-the-line voting, with a maximum 25 per cent chance of a swap in seats.

These results can be understood in terms of the law of large numbers, since there are many more voters than parties. That is, for below-the-line voting, the random individual preferences are distributed quite evenly at each round.

This is in contrast to above-the-line votes, which are also transferred between parties randomly but en masse. The minor party voters thus take on different roles in each case – either their first preferences enable a minority to succeed, or their random additional preferences decrease the likelihood of this happening.

Similar roles have been explored in the collective decision-making of animal groups, where having lots of uninformed (randomly voting) individuals can either help or hinder the ambitions of a few, depending on how decisions are made. It is doubtful, though, that ants spend as much time as we do agonising over decisions.

These statistics are interesting for pundits or analysts or political scientists, but what's in it for the voter? One counter-intuitive conclusion is that voters for minor parties are more likely to see their first choice win a seat if they take the easy option and vote above the line.

This is somewhat ironic, since voters for minor parties are more likely to vote below the line. While doing so will better reflect their individual preferences, it is unlikely that these votes are similar enough to win an election.

There is a catch, though. If a voter stays above the line, the increased chance of one minor party winning a seat comes at the cost of not knowing which one. Whether it's worth taking this risk is for the voter to decide.

Mathematicians have demonstrated that there is no such thing as a perfect voting system. This makes it difficult and contentious to pick a suitable system on which to base a democracy. Often the result is complex and intimidating.

But mathematical analysis and simulation, while perhaps not as straightforward as the line for the sausage sizzle, can at least make these complexities easier to navigate.

Alex Szorkovszky is a researcher in the Collective Behaviour group at Uppsala University in Sweden.

References

Nicholas R. Miller. The butterfly effect under STV. Electoral Studies 26, pp. 503–506 (2007)

Iain D. Couzin, Jens Krause, Nigel R. Franks and Simon A. Levin, Effective leadership and decision-making in animal groups on the move. Nature 433, pp. 513-516 (2005)

Nigel R. Franks, Stephen C. Pratt, Eamonn B. Mallon, Nicholas F. Britton and David J. T. Sumpter, Information flow, opinion polling and collective intelligence in house–hunting social insects. Phil. Trans. R. Soc. Lond. B 357 no. 1427 pp. 1567-1583 (2002)

K.J. Arrow, A Difficulty in the Concept of Social Welfare. Journal of Political Economy 58(4), pp. 328–346 (1950)

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