Why the parliament expresses the “will of the people” better than any of its MPs including its PM and better than the brexit referendum

In 1907 Francis Galton published a paper in Nature titled Vox Populi. In it he discusses how the collective judgement of the crowd is better than that of the average individual (a fact which goes back to a simple mathematical identity, also known as diversity prediction theorem, which states that the squared error of the average of all predictions is smaller than or equal to the average squared error of all prediction because from the latter we need to subtract the diversity of the individual predictions to get the former). In what we would, today, call policy recommendations of his paper, Galton states:

The result is, I think, more creditable to the trustworthiness of a democratic judgement than might have been expected.

On the other hand, consider as Marquis de Condorset might have done in the 18th century, three voters Peter, Paul and Mary ordering three different alternatives, let’s say: Remain, Deal Brexit and No Deal Brexit. Here is how their preferences might be:

Peter: Remain, Deal Brexit, No Deal Brexit

Paul: Deal Brexit, No Deal Brexit, Remain

Mary: No Deal Brexit, Remain, Deal Brexit

All three are opinionated “patriots” who want “the best” for the UK. Peter wants to remain but if not feasible he wants an amicable divorce. Paul wants a divorce but if possible an amicable one and Mary wants no compromises: either leave “proud” or don’t try it at all.

It is irrelevant who is “right” or “wrong” but we can assume whatever contributes to the opinion of each individual does so consistently. In other words they might change their minds but each time you ask them they have an ordering which puts the three options in order. Their personality, gender, hormonal levels, socioeconomic status, childhood etc all might play a role. Now let’s examine which option has majority. Remain is preferred over Deal Brexit by a 2/3 majority (Peter and Mary), Deal Brexit is preferred over No Deal Brexit by a 2/3 majority (Peter and Paul), No Deal Brexit is preferred over Remain by a 2/3 majority (Paul and Mary). So the “democratic process” picks Remain over Deal, Deal over No Deal (hence Remain is better than No Deal) but it also prefers No Deal over Remain. So it prefers Remain over No Deal and No Deal over Remain!

So what gives? The mess the UK is in has some simple mathematics which if paid attention to could help bring emotions down and make some very simple thoughts which might lead to some very reasonable decisions. A referendum is too crude a tool to use to make important decisions especially with a turnout of 72% and a Leave to Remain outcome of 51.89% to 48.11%. Marquis de Condorcet will tell you that when the Leavers say that the referendum is “the will of the people” they are just bullshitting the public opinion. If you posed seemingly equivalent formualtions of the same question you could very well get contradictory outcomes such as leaving is better than remaining and remaining is better than leaving. On the other hand Galton will tell you that the parliament as a collective has a better chance at figuring things out than any MP (or citizen) including the PM.