Are there too many options in the Brexit vote?

The Frenchman and child’s game which provide an explanation of our Brexit dilemma and why the Prime Minister has an impossible task.

Article updated: 26 March 2019 11:00am Author: Richard Stone

It is perfectly understandable if the Marquis de Condorcet is not a name which is readily recalled from school day history lessons. However, this 18th Century French philosopher and mathematician wrote extensively on social theory and in 1785 he wrote on the issue of probabilities and majority voting. His work set out what became known as Condorcet’s paradox – and it is this which, over 200 years later and now in an era of mass emancipation, captures exactly the essence of the dilemma over Brexit.

What Condorcet set out in his work was the fact that when presented with binary choices voters may express preferences which appear contradictory. So, if given a choice between candidates A and B a majority may prefer candidate B. When given a choice between B and C they may choose C. Logically therefore when given a choice between A and C they would choose C, however, in reality they choose A – therefore creating a paradox.

This is easily captured in the concept of the child’s game “Rock, Paper, Scissors” – each option has the ability to defeat the others in a manner which is cyclic. No single option can command victory against all other options when presented in a binary manner.

In politics and elections this happens when the electorate is being presented with a binary choice but one side is effectively capturing a range of other outcomes.

Brexit means different things to different people

When set in the context of the Brexit vote the issues this creates become clear. In the original referendum in 2016 voters were given a choice between Remain and Leave. Although it may be debated what our future would look like, if we chose Remain there was a clear and widely accepted view of what that entailed. Leave on the other hand was ill defined and meant many different things to many different people. For some it meant staying close to the EU (for example, in a customs union and perhaps even still in the single market) but outside the main institutions; for others it meant leaving abruptly and without a deal. For some it was about containing immigration; for others about being able to agree international trade deals. While for some it was just an opportunity to reject the status quo. This was all best captured by the phrase “Brexit means Brexit” which actually defined and said nothing.

In essence Leave won precisely because it brought together a wide range of opinion which, when collected together, could defeat Remain but which had no collectively agreed view of what it was actually ‘for’.

Now as the Prime Minister has tried to get her deal through Parliament we see the same issue arising again. Binary choices being given where only one side of the choice is clearly defined and the wide range of opposing views – which disagree with themselves – create a majority against. Therefore when the Prime Minister’s deal was put before Parliament, the European Research Group, which favours a harder Brexit or indeed leaving with no deal at all, walks through the same lobbies as those fundamentally committed to remaining in the EU, or to negotiating a softer Brexit (the official Labour Party position). Those opposed to the Prime Minister’s deal do not themselves agree other than in so far as they agree they don’t support her deal – for a wide range of very different reasons.

Whatever shape the deal was that the Prime Minister agreed, it would always have struggled – indeed likely failed – to get Parliamentary support. As any deal would only serve to unite opponents with widely differing views. In short Theresa May has always faced an impossible task.

Too many options

Forming a majority for any one option when put to a vote is, in this context, impossible. Every option pulls together a host of opposing views sufficient to prevent a majority in favour of anything. The only thing MPs clearly support is that ‘no deal’ should not happen – but that is relatively simple to command an overwhelming majority for as, in itself, it is not a vote ‘for’ anything and enables those supporting all other options and ranges of deals through to staying in the EU to coalesce against a ‘no deal’ outcome.

There are a number of voting systems and methods which have been proposed to address the paradox which Condorcet identified. For example, running various paired elections between different options or getting voters to place the options in their order of preference. These are complex and to apply those to a referendum type scenario across the population would be unmanageable.

The only solution I can conceive in the current parliamentary impasse – an impasse which Condorcet’s paradox suggests is unlikely to be broken – is to return to the people with a genuinely binary choice. This in itself could have various iterations.


It could be “Remain” vs “the agreed deal”. Alternatively it could be “Leave on the agreed terms” vs “Leave with no deal” – on the basis that the population has already voted to leave. This highlights the challenge of trying to boil this down to a straightforward binary choice.

Is there a solution to the paradox?

Condorcet does not provide us with a solution but over 200 years ago he identified the issue democracy can present and highlights the issue of trying to secure a majority for a single option among a range of possible outcomes. Where those supporting the different outcomes do so with passion and vigour making compromise all but impossible the Prime Minister arguably has an unachievable task ahead of her.

At some point the Government and the Prime Minister must boil the choice down to two clear binary options and get Parliament to then support putting the choice between those two outcomes to the people where each of the two options is a vote ‘for’ something clearly defined.


These views are those of the author alone and do not necessarily reflect the view of The Share Centre as a whole.

Richard Stone portrait photo
Richard Stone

Chief Executive

Richard is a qualified chartered accountant who has held several director roles across the financial services sector. His responsibilities include all aspects of oversight, including the group's strategy for growth, and encompass control and management of the group's business.

See what else we have to say