The trouble with referendums

How a democratic process can lead to consequences of a questionable democratic character

Koen Smets
Sep 5, 2016 · 8 min read

People are a motley crew, really. Every one of us has their own set of preferences. Thankfully the free market makes sure that someone offers exactly what we want — no matter how exotic or how obscure. We choose these products over here and those services over there, and thanks to the miracle of the market all these preferences are aggregated, and suppliers do whatever is needed to match their offer with what we consumers expect.

In a way, a democratic electoral system has a similar purpose: to combine the preferences of all the voters, and produce policies that represent the wishes of the entire electorate.

Unfortunately, elections don’t always deliver the goods (or services) in the way the market does.

To understand why, it’s a good idea to start with conventional micro-economics, which describes how rational individual actors take decisions. Behavioural economics may indeed have one or two things to say about this assumed rationality, but there are some important characteristics of rational agents that us real humans really pretty much possess. One of these is the so-called transitivity of preferences. This means that, if you prefer apples over oranges, and oranges over pears, you also prefer apples over pears. Seems logical and self-evident.

A little intransitive

Of course there are situations where that doesn’t quite apply. Imagine that the barista in your local coffee purveyor proposes, along with to the normal Americano (let’s call it “A0” on the occasion) for £2.50, a slightly weaker version, “A1”, for £2.45 — with a little less coffee and a little more water. Turns out you can’t taste the difference. So why pay more, even if it’s only 5p? The next day, he has a new proposition: there is now also a slightly more diluted variant, “A2”, costing only £2.40. Again you do not taste the difference, and you expect to save yourself 10p in the process.

But the barista also lets you compare the taste of this newest variation with that of the original. Not only do you notice a difference in taste, you also think the A0 tastes so much better than A2 that you’re prepared to pay the full price of £2.50 for it rather than go for the cheap option. And now your preferences violate the transitivity axiom: you prefer A2 over A1, A1 over A0, but also A0 over A2.

This is surely an intriguing form of irrationality, but in practice such artificial situations don’t really occur that often, and as individuals we mostly act as if our preferences are transitive.

The fact that we don’t all have the same preferences — Jane thinks apples > oranges > pears, but for Peter pears > apples > oranges is not a problem for a well-functioning market. It will make sure that each type of fruit is available in the required quantities, as they can exist alongside each other.

But what if there is a presidential election with three candidates? Assuming every voter perfectly follows the transitivity axiom, will the electorate as a whole then also be transitive in its preferences? In other words, if in a run-off between candidates A and B the winner is A, and in a head-to-head between candidates B and C victory goes to B, then in a contest between A and C, A will be elected.

Condorcet’s insight

French mathematician and political scientist Marquis de Condorcet examined this problem in the late 17th century, and concluded that an electorate that consisted of rational voters could very well have intransitive preferences. You might think this is only a problem for theoretical psephologists, but it is quite likely, as among others Jonathan Portes suggested, that this is relevant to the EU-referendum in the UK on 23 June 2016. What happens if we work this out with some actual numbers?

Let’s look at three relevant options in the debate: (1) full EU-membership (the status quo); (2) the Single Market option (also sometimes referred to as the EEA/EFTA or Norway option: membership of the single market and the freedom to establish trade treaties with third countries, but also free movement of people and a contribution to the EU coffers); and (3) full sovereignty (complete control over immigration, no contributions to the EU, and no membership of the single market).

Officially, the ballot paper did not specify whether ‘Leave’ meant option 2 or option 3, but considering the importance of migration and taking back control in the campaign, and the statements by Vote Leave leaders rejecting the Norway option, it is not unreasonable to assume the de facto choice in the referendum was between options 1 and 3. As is well known, ‘Leave’ won by 51.9% versus 48.1%. But this clear win conceals the complex reasoning of the diverse voters, all with their rational, transitive preferences.

Look at the Remain voters, for example. What would their preference be if they had to choose between options 2 and 3? Some of them would prefer the Single Market option (because they consider it the next best thing to EU membership), but others might go for option 3 — “if we have to leave the EU, we might as well really take control and be master of our own destiny.” It is also quite possible that some Remain voters would have chosen for the Single Market option, had the referendum been a choice between option 1 and option 2.

And we can look at the Leave voters in the same way: free traders would prefer option 2 over option 3, while people who want to curb immigration would prefer option 3. In a contest between options 1 and 2, some of them might be of the opinion that, if there’s not going to be a stop to migration, the country might as well just stay in the EU, as the cost and complexity of realizing option 2 is too high for no real benefits in their eyes.

The three options can be ordered in six different ways, and so we can define six groups of voters according to their ranked preferences:

Groups A-C together are the ‘Remainers’ representing 48% of the electorate. The opinion polls since the referendum suggest that most of them are genuine Europhiles (A). Alongside is a category of voter (B) that prefers the status quo, but if that is not available, would rather be totally sovereign than in the single market with its restrictions and costs. A third class © would ideally like a looser relationship with the EU, but judges that staying within the EU is the best way to achieve that (like the Adam Smith Institute’s executive director, Sam Bowman).

Opinion polls would also support the assumption that a majority of the Brexiteers would have the Sovereignty option, with full control of immigration, as their first preference (groups E-F). Within that segment there are those (E) who would, if there is not to be any control of immigration, prefer EU membership over single market membership, as the latter is seen as an expensive way of getting none of the relevant advantages of being out of the EU. Finally, there is a class of Leave voter (D) that has the Single Market option as their first preference, and dislikes EU-membership the most.

Other referendums

This distribution of voting preferences is inevitably speculative, but it is not implausible. What would the result be of a referendum with a different binary choice? Imagine Brexit was a fact — “Brexit means Brexit” as it were — and the status quo option was not available. Groups A, C and D would then go for the Single Market option and B, E and F for the Sovereignty option, giving the single market the edge with 56% vs 44% — a rather larger margin than in the actual referendum.

And if the choice is between remaining in the EU and the Single Market option, groups A, B and E would have gone for the status quo, with a total of 57% vs 43% — an even larger margin.

The intransitivity is clear (and exists for a wide range of percentages): the electorate chooses EU membership over the Single Market option, the Single Market over the Sovereignty option, and the Sovereignty option over EU membership. Under this hypothesis, it also emerges that the referendum as it was held produces the smallest margin between two options of the three possible binary referendums.

That raises questions about democratic legitimacy of a referendum that gives a choice between two out of three possible options. As the good Marquis noted, it is very well possible that binary choices between three possible options do not produce a strict transitive preference. There simply is no absolute winning option.

Is there another way to determine the people’s preference? We could look at how many people have each of the options as their first preference. For EU membership, that would be 41% (A and B), for the Single Market option 30% (C and D) and for the Sovereignty option 29% (E and F). The idea that the latter option does not carry so much weight is supported by a recent Yougov poll, suggesting more than half of the British would be happy to accept free movement of people in return for a Free Trade deal:

So what is the problem?

The figures in this analysis are of course hypothetical. But they are (a) entirely in line with the actual referendum result, and (b) compatible with the various opinion polls that have been conducted since 23 June.

The point of the analysis is not to claim that they represent the reality. Its point is to show the shortcomings of an election in which a binary choice is offered between a subset of possible options, and in particular to show that it is unsafe to conclude from a binary referendum that the winning option represents the preference of a majority of the electorate among all options, or indeed that it represents the option that has the most support among the population.

The analysis presented here holds irrespective of what we call options 1, 2 and 3. It demonstrates that an option gaining 52% of the popular vote in a single binary referendum may well be the least popular option, based on the voters’ preferences.

In practice, with what we know, there is a plausible reality behind the referendum outcome that would be at odds with a current rhetoric about a ‘hard Brexit’. More importantly, should the Prime Minister eventually put forward a proposal in which Single Market membership is rejected in order to ‘take back control’ of the UK’s borders, then she might very well be rejecting the option with the strongest democratic support in favour of the option with the least support, and give in to the smallest, but most vocal minority.

Referendums have the reputation of being the ultra-democratic — many people believe that they do not suffer from the (unquestionable) flaws of parliamentary democracy. But they don’t deserve this reputation. As de Condorcet established, they are structurally incapable of delivering a robust picture of the true preferences of the electorate in all but the simplest binary choices. They should not be used to justify momentous — or indeed any decisions — in the name of democracy.

(An earlier version of this article was first published in Dutch at

Originally published at on September 5, 2016.

#NoDust on Brexit

Posts relating to the questions surrounding Britain's relation with the European Union

Koen Smets

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Accidental behavioural economist in search of wisdom. Uses insights from (behavioural) economics in organization development. On Twitter as @koenfucius

#NoDust on Brexit

Posts relating to the questions surrounding Britain's relation with the European Union

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