What makes a majority?

When is a majority large enough to be considered decisive?

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In May this year, Muirfield golf club made the headlines after voting to not accept women as members and keep it a private men-only golf club. In a radio interview, golf commentator Peter Allis did point out that wives and partners can already play as guests and visitors:

They get all the facilities. If [a woman] wants to join, you’d better get married to somebody who’s a member’

Charming. Whilst the attitude towards women may seem outdated, the club’s voting method demonstrated more consideration than another vote held this year. They decided that a vote to introduce change should require a two-thirds majority to be decisive.

A similar approach applies to science. For a trial to be considered successful, it has to pass a significance test. Deciding what is or is not significant is a debate in its own right. To keep it simple, imagine you were assigned to one of two groups in a trial for a drug claimed to cure an illness that everyone involved in the trial suffers from. You don’t know if you are in the group receiving the drug or the group receiving the neutral placebo. The number of people with improved health following the trial were recorded as follows:

How confident would you be about knowing whether you received the drug or the placebo? When comparing test groups and control groups, if the results are close then the claim that the drug made a difference will likely be rejected.

In criminal trials, even two-thirds is often insufficient to be considered a majority verdict. Depending on the seriousness of the crime and the sentence that will be handed out, the judge may require a unanimous verdict from the jury. If they fail to reach one, then a weaker verdict will sometimes be accepted but still well above two-thirds. In a civilised society, we consider it preferable to let bad people get away with their deeds than mistakenly lock up good people. Still happens, but that’s a different story…

The reason for this ramble through percentages is the EU referendum held in the UK. For reasons best known to the government, they decided in December 2015 that there would be no threshold for the result. Even if just 1 vote separated the two choices, creating a majority of 50.0000015%, it would be accepted as decisive no matter how disruptive such a close outcome would be. This was despite the government having experienced just such a situation 12 months earlier when residents in Scotland were given the option to stay or leave the UK. 55.4% voted to stay and 44.7% voted to leave. The closeness of the vote ensured immediate talk of another referendum.

The following map shows the results of the EU referendum plotted by area within Great Britain*:

You would think from this map that the verdict was very clear – excluding Scotland and London, people voting overwhelmingly in favour of leaving the EU. But whenever a politician states that the UK public voted decisively in favour of leaving the EU, they are stating that 51.89% voted to leave and 48.11% voted to stay. To see how close the vote was, here are the results presented in two ways. On the left, are the votes counted to remain in or leave the EU. On the right is the full set of outcomes – those who voted to remain, those who voted to leave, those who spoiled their ballot paper (ticked none or both boxes), and those on the electoral register who didn’t vote:

Just considering those who managed to successfully tick one box, and only one box, on a piece of paper, the result was close to 50:50. When considering everyone eligible to vote, the result was one-third wanting to leave, one-third wanting to stay and one-third didn’t care enough to vote either way. I would not consider this to be a decisive outcome.

If the EU referendum had followed the example of Muirfield golf club, and required a two-thirds majority for the vote to be decisive either way, this is how the results would have looked when plotted by area:

If a two-thirds majority had been required, just 10% of areas would have voted to leave, 6% would have voted to stay in the EU and 84% would have failed to reach a decisive verdict. If a weaker threshold had been set of 60%, it still would have meant nearly two-thirds of areas would have failed to reach a majority (62.04%). Even setting the threshold as low as 52% would have resulted in over 13% of areas not reaching a majority. The following image shows the different thresholds:

Different tests require different majority verdicts. It is rare to require a unanimous decision and when a decision has to be made one way or the other, even the smallest of margins will be accepted. But when the vote is to put in place a long-term process that will not easily be undone, I’d rather see a method that improves on tossing a coin to decide the future.

A final note, in case anyone starts ranting for or against Brexit. This post isn’t about the direction of the outcome. 52:48 was a bad result whether it was to leave or remain. It doesn’t mean half of the people got the vote right and half got the vote wrong. It means a significant number of people didn’t know or care which way to vote.

* Apologies to Northern Ireland – my shape file for plotting maps is GB-only. And apologies to the handful of remoter islands north and west of Scotland cropped out of the map for convenience.

References

• Muirfield gold club votes not to accept women members – FT.com, 19 May 2016
• Peter Allis on Muirfield’s women membership vote – BBC Radio 5 live, 19 May 2016
• Scottish independence referendum – BBC News, page as of 22 August 2016
• Why referendums should be banned – Paul Evans, 13 December 2010
• Government response petition for a second EU referendum – petition.parliament.uk

Election results courtesy of the Electoral Commission

Spatial map boundary lines courtesy of Ordnance Survey open data

Related posts

• When third place matters most

Featured image: ‘man casting his vote‘ – Getty images via Flickr, kindly shared under Creative Commons license, found via Photopin

Still here? Thanks for reading! Here’s one final map, showing how each area voted as a percentage. Strong red means Leave, Strong blue means Remain, the more faded the colour, the closer the result was to 50:50: