Designing a ‘people’s vote’ is a political minefield all on its own

It’s like a giant jigsaw puzzle where, however hard you try, the pieces will never fit together.

That’s the state of play with Brexit. Whichever way you present the options, nothing seems to command a parliamentary majority.

Amid the chaos, talk has increased of a so-called “people’s vote” as the only way to break the impasse. But those who believe this to be the easy solution are sadly mistaken.

What has been floated so far is a three-way vote between the deal on offer, no deal at all, or staying in the EU. The question that arises is which method of counting the votes one would use. There are three options.

The first is a simple plurality vote: whichever option gets the most votes wins. This has the advantage of simplicity, but the disadvantage that, say in a 40:30:30 split, the winning option does not have the support of the majority.

The second is an instant run-off, whereby the lowest scoring option is eliminated and the votes from that choice are allocated to the second preference of those voters.

This is reasonable, except that it tends to disadvantage compromise, since people are unlikely to put any compromise as their first preference.

Finally, there’s the “Condorcet” method: each option is judged against the other options. If the deal beats no deal, remain beats no deal, and the deal beats remain, then the deal wins.

But this approach can result in a circle where there is no option that ends up beating all the others.

Professor Christoph Borger of Tufts University has suggested that, in those cases, one then falls back on a points system: a first preference vote would count for three points, a second for two points, and a third for one. The winner is the option with the highest total score.

However, this system only works if those ballots that do not contain preferences for all three options are completely discounted. Otherwise people would only vote for their preferred option to deny the other choices any points at all.

Crucially, several polling companies have recently surveyed voters and analysed the results using the various methods above. They have shown that the result is likely different under each of the options. Which voting method is chosen could therefore be a major factor in influencing the winner.

In 1972, Kenneth Arrow won a Nobel Prize in economics for his work on social choice theory. What is known as Arrow’s impossibility theorem states that, when voters have three or more options, no ranked voting system can convert voting preferences into a community-wide ranking while still satisfying democratic fairness criteria.

In other words, all voting systems are highly imperfect.

But then again, is it fair to argue that a two-way referendum, with the winning side taking 52 per cent of the vote, reflects community-wide preference?

Or, to put it another way, in our understanding of the principles of liberal democracy, can ignoring the expressed wishes of 48 per cent of the electorate really be considered “the will of the people”? Can such an approach ever promote a cohesive society, or is the consequent damaging polarisation inevitable?

There is no perfect system. A people’s vote may well be the only way forward in the event of parliamentary gridlock, but those who believe that it will be a straightforward process are deeply misguided. It will unleash the mother of all battles around how the referendum is structured, and each faction will want the counting system most likely to favour their own particular preference.

That jigsaw may well be just as insoluble as the one we currently find ourselves facing.