Referendum and Overlap

I have seen a few referendums in the past:

• The Turkish presidential system referendum with 51.41% for the new system. Source
• The Brexit with 51.89% for leaving the EU. Source
• The Irish abortion referendum with 68% for allowing it. Source
• The Swiss "Billag" vote for publicly financed broadcasting with 71.6% for keeping the current system. Source
• The annexion referendum on the Crimea with 95.5% to join Russia. Source.

The first two ones are awfully close to a tie, and that makes it problematic. When the result of a "Yes or No" referendum end that close to 50%, I would argue that the referendum asks the wrong question.

Say for instance the opinions about a particular topic can be expressed on a bounded one-dimensional scale from 0 to 100. This of course is a vast simplification. Then we also assume that these numbers are the actual opinions. Let's assume they are distributed like this:

Now when one has to vote in a "Yes or No" referendum, people above 50% will vote "Yes", the others "No". For this particular distribution, the vote will end like this:

I think of this more as a psychological study or a physics experiment where we want to find out the distribution of opinions of the population. Basically I want to have this histogram that I have showed above. But we have to ask the people some questions in order to find out where they stand. If we just prompt with "Yes or No", then we basically measure with the function on the left:

But the distribution in the population would be much better covered by a triangle question.

I happened to be in Dublin when the abortion campaign was going on and got to see various posters. The things written on them usually were on the extreme. And both variants seem absurd to me. If the "morning-after pill" is illegal and abortion is illegal, what do you do with contraception failures? Force the teenagers to carry out the child and prevent them from finishing school properly? What about rape victims?

The "No" side was arguing that in case the law got changed, babies could be aborted right up until birth. Which is also nonsensical, but a very emotional argument.

So I think that people should meet in between. But once there is the referendum, it is either a hard "Yes" or a hard "No". People in the middle will become polarized or do not vote at all because they feel torn. My impression is that once the political negotiations about something start to stall, both sides do a tactical retreat and then meet later in the referendum. It feels a bit like a war.

And the worst part is that once the vote has passed with 52% percent, there will be 48% percent of the people rather unhappy. If rather the question was "Do you want to allow abortion with limitations and allow morning-after pills to reduce the need for them in the first place?", as visualized with the triangle shape, most of the people would be happy and extreme opinions on either side would be unsatisfied.

Similarly with the Brexit. They only had two extremes to choose from, and by chance it got to become the Brexit.

Therefore I start to dislike referendums. At the core they measure the difference between "Yes" and "No". And for a wide deviation within the population, there will be a relatively large statistical error on this. Of course the vote is not just a representative study but gives everyone the opportunity to be heard. So technically there is no uncertainty on the result.

It would seem to make sense to have some sort of overlap computed. For this one would have to measure the distribution of opinions in the population more carefully. A simple way to ensure that the referendum does not make a significant fraction of the people unhappy is to demand that the vote has to be at least 70%/30% or more extreme in order to be implemented. If it is more equally distributed, the referendum asked the wrong question and the politicians need to get in touch with the people again.