Brexit Referendum: An Incomplete Verdict

C Penny Tridimas[1] and George Tridimas[2]

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The purpose of this paper is to use the insights of collective choice theory to examine the format and outcome of the Brexit Referendum. It is argued, first that the question asked by the Referendum was incomplete for it failed to address the choice between “Hard” and “Soft” Brexit resulting in a deficiency of legitimacy. Second, this observation gives rise to the question of the actor who can remedy the deficiency, either the electorate in an additional referendum on the form of Brexit, or Parliament in an active role in negotiations.

The Brexit Referendum


The decision of the UK electorate to leave the EU following the “In-Out” Referendum of 23rd June 2016 has led to a debate regarding the terms of future relationship between the UK and the EU. Specifically, the debate can be summarised as a conflict between “Hard” and “Soft” Brexit.

“A ‘soft Brexit’ is commonly taken to mean continued Single Market membership, including free movement of not only goods, but also services, capital and labour- as the non-EU states Norway, Iceland and Liechtenstein have. The hardest ‘hard Brexit’ is usually understood to mean the UK having no preferential relationship with the Single Market and relying only upon World Trade Organisation (WTO) rules. This implies not merely non-membership of the Single Market (meaning the potential for non tariff barriers to be adopted, hindering trade) but also the imposition of tariffs on at least some trade in goods between the UK and the EU.” (Political Studies Association, 2016)

The question put to the UK electorate was “Should the United Kingdom remain a member of the European Union or leave the European Union?”. 52% of the electorate voted to leave the EU, while 48% chose to remain a member. It is clear that the “Leave” answer did not specify what the future relationship with the EU would be. It is the contention of this paper that had the two options regarding the nature of that relationship been included, the referendum result would have been radically different. Further, and perhaps more controversially, the absence of such a choice lends no democratic legitimacy to either “Hard” or “Soft” Brexit.


Referendums are thought to provide policy outcomes that reveal the preferences of the majority of voters. The reason is that when voters are perfectly informed about the issue at hand, they circumvent the principal-agent problem that may bedevil the relationship between constituency voters and their MPs, where the discretion afforded to the latter may be abused to the detriment of the interests of the former.[1] However, the paper argues that the Brexit Referendum did not fully benefit from these advantages. Judging from the accusations that each side hurled to the opposition regarding scenarios of economic meltdown and unsubstantiated claims of the costs of EU membership, it is debatable whether voters were perfectly informed.


A further reason for an incumbent to call a referendum is to resolve questions which cut across partisan lines where the governing party and / or the opposition are split on the issue[2], like constitutional questions regarding sovereignty, e.g. Scottish independence, and of course UK membership of the EU. Since 1972, the date of UK’s entry to the European Economic Community, both the Conservative and the Labour parties have been split on the issue of European Union membership. The presence of intra-party rifts in the 2015 Conservative Government is corroborated by the fact that the then Prime Minister, David Cameron, was forced to allow his party, MPs and Cabinet to campaign on different sides of the Referendum question, instead of imposing party discipline in support of his declared position of “Remain”.


Theoretical considerations of voter decision-making


The Brexit Referendum can be modelled as a sequential game of two stages. In the first stage, there are two possible outcomes, “Remain” and “Leave”. If the result is “Remain” the game ends. If the result is “Leave” then a second stage follows, where the options are “Hard” and “Soft” Brexit. However, as things are now, it is not clear whether the second stage will be decided by the electorate upon holding a second referendum, by the Prime Minister, or by Parliament. In addition, the uncertainty regarding the second stage outcome implies that at the beginning of the game, that is when the “In-Out” Referendum was held, voters were imperfectly informed about the settlement following an “Out” result. This in turn leaves open the possibility that a different outcome might have prevailed were the voters fully informed that a “Hard” rather than a “Soft” exit was the likeliest outcome. These considerations imply two lines of inquiry, first, the type of settlement with the EU, and second, the actor who approves the settlement.


Figure 1 graphically shows the Referendum outcome as a two-stage game. The top node, “Remain”, shows the outcome and percentage of votes polled, and indicates that the voting game reaches an end where no other actions are taken. The bottom node, “Leave”, is followed by two different outcomes. The upper node shows an outcome of “Hard Brexit” receiving  of the popular vote; the lower node shows an outcome of “Soft Brexit” receiving of the popular vote. With It is therefore unclear how much electoral support each of the latter two options received.

More specifically, since the “Leave” option received of the vote, a rather small fraction above the required majority threshold of votes, one may reasonably assume that neither the “Soft” nor the “Hard” exit options commanded the majority. In fact, it is more likely that they both polled less than the received by the “Remain” option. It may however be more difficult to establish whether the “Hard” or the “Soft” option was more popular than the other. As a result, we can represent the Referendum outcome in descending order of votes asor. In light of these observations, the following two interrelated inferences are derived regarding the format of the referendum and the actor responsible for negotiating the final settlement between the UK and the EU.


Implications for format


Had the Referendum offered a three-way choice instead of a binary choice, and a rule of first-past-the-post applied, the UK would still be a member of the EU. This rests on three arguments. First, the incompleteness of the binary nature of referendums; second, the validity of Independence of Irrelevant Alternatives; and third, the type of majority rule applied. With respect to the first, we note that since “Leave” is only an intermediate stage, offering three choices each representing a final outcome, a three-question Referendum is justified. Further, a three-question Referendum would have provided a more accurate recording of the majority preferences of UK voters. Of course, there is no international precedent of a three-question referendum. Nevertheless, an “In-Out” EU membership referendum was unprecedented, so changing the question format would not have been as outlandish as it may first appear to be.


Evidence that a three-question referendum would have offered a more accurate picture of popular preferences comes from the split of the Euro-sceptic camp into two groups, the official “Vote Leave” Campaign headed by Conservative Ministers Michael Gove and Boris Johnson, and the “Leave.EU” Campaign headed by the Euro-sceptic MEP and then-leader of UKIP Nigel Farage. The former is presumed to have been more sympathetic to the “Soft” Brexit option than the latter. It follows that a three-question referendum may have improved the information available to the voters. Arguably the presence of two Campaigns on the “Leave” side obfuscated information and might have erroneously led some voters to vote as if they were deciding the second, not the first, stage of the game.


Second, for “Remain” to win a three-question Referendum, that is, “Remain” still receivessupport using a first-past-the-post voting system, the Independence of Irrelevant Alternatives condition must not be violated.[3] The condition requires that despite the presence of two alternatives for “Leave”, those in favour of “Remain” still regard “Remain” as the most preferable option. In other words, full EU membership is preferred to any other relationship with Europe presented in the Referendum.


Third, with respect to the type of majority rule, if assumptions regarding the constellation of voter preferences as above are correct, then there is no doubt that use of first-past-the-post would have rejected “Leave”. This conclusion illustrates the tension between the simple-majority voting rule and first-past-the-post. Further evidence of the presence of the tension comes from comparing the voting outcomes of the constituent nations of the UK. The “Out” option was supported by 53% and 52% of the English and Welsh electorate respectively. On the contrary, “Out” received only 38% and 42% support in (the less populous) Scotland and Northern Ireland correspondingly. Arguably, applying a voting rule that would have accounted for the varying preferences of the constituent nations may have strengthened the legitimacy of the outcome and reduced constitutional strains relating to the devolved settlements of the UK.


First-past-the-post is not the only electoral rule that produces a single majority outcome required for the type of choice examined (proportional representation systems cannot be used in referendums because they do not generate a clear winner). Using the Alternative Vote system would have also produced clear-cut results.[4] However, the actual outcome is not predictable. The reason is that the Referendum did not record a full ranking of voter preferences (that is, how each voter ranked “Remain”, “Soft” and “Hard” Brexit). Hence, we do not know which alternative is eliminated and how votes are transferred to the rest. This means that it is not possible to establish what the outcome of a three-question Referendum using the AV system would be.


Implications for actor


Our previous analysis identified a serious deficiency of the Referendum process: the Referendum has not produced a democratic mandate for either “Hard” or “Soft” Brexit. To determine which of the two commands greater popular support, either a second referendum is needed to legitimise the negotiated relationship with the EU, or the House of Commons must be actively involved in the objective of the negotiations. The latter route begs the question of whether the sitting Parliament voted in 2015 before the Referendum or a new Parliament with a fresh mandate votes on the objective of the negotiation. Irrespective of whether a general election is called, the analysis implies that MPs have the authority to remedy the above deficiency, since they are representatives of their constituents. This is in accordance with the principle of Parliamentary Sovereignty enshrined in the UK Constitution. Henceforth, due to the ambiguity regarding preferences for “Hard” or “Soft” Brexit, and contrary to the benefit of referendums to sidestep the problems of the principal-agent relationship, the UK Parliament must play an active role in the negotiation of the UK’s relationship with the EU.


In R (Miller) v Secretary of State for Exiting the European Union, the Supreme Court held that the Prime Minister does not have the prerogative power to trigger Article 50 TEU (which provides the legal process for a Member State to follow if it wishes to leave the EU), but must seek Parliamentary authorisation. Our analysis suggests that regardless of whether such power rests with the Prime Minister, and in the absence of an additional referendum, the involvement of Parliament in the process would enhance democratic accountability and strengthen the legitimacy of the final outcome of the negotiations.




This paper argued that the question asked in the Brexit referendum was incomplete. The “Leave” verdict begs the question of what relationship the UK will have with the EU, which suggests that a complete set of questions should have included three options, “Remain”, “Soft” or “Hard” Brexit. In view of the small margin of votes by which the “Leave” option won, we advance two arguments. First, “Leave” would not have won if a first-past-the-post rule over had been applied over the three options. Second, since the division of the support of “Leave” between “Hard” and “Soft” is not known, either a second referendum is required to determine the preferred settlement, or Parliament must be involved in the decision making process.





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Electoral Reform Society. 2011. Introducing the Alternative Vote. Available from, accessed 14/01/17

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Nurmi, H. 1997. Compound majority paradoxes and proportional representation., European Journal of Political Economy, 13, 443-454

Political Studies Association. 2016. Brexit and Beyond How the United Kingdom Might Leave the European Union. A Report. Available from, accessed 02/11/16

Tridimas, G. (2007). Ratification through referendum or parliamentary vote: When to call a non required referendum? European Journal of Political Economy, 23, 674–692.



1 Comment

  1. Interesting analysis of where incomplete information (or badly-formed questions) leave the referendum. I think your conclusion though is a little too simplistic, however: just as there were shades of ‘Leave’ (soft & hard) which emerged only after the first decision, necessitating a further choice, so it should be assumed that ‘Remain’ was also not a simple choice. Cameron had set out to renegotiate some aspects of ‘Remain’ (and largely failed) but there was considerable dissatisfaction with the status quo.

    This argues that the ‘game’ is a more complex one, where the Remain ‘bloc’ was also split on what remaining meant – status quo, or renegotiation. Although from where we are now this might not seem significant, it seems to me that a Remain vote would similarly have generated ‘hard’ and ‘soft’ camps very quickly, based on the levels of dissatisfaction that led to the ‘Leave’ vote. In other words, you assume that only the ‘winners’ had a range of views, while all the ‘losers’ were a united bloc.

    With ‘hard’ and ‘soft’ Remain options, too, the voting game becomes more complex, with no obvious winners. And why only ‘hard’ and ‘soft’? Each in fact is a continuum… In fact, the ‘perfect’ referendum would only be one that set out every choice from ‘hard’ Remain to ‘hard’ Leave as you remove one facet then another. It would be interesting to guess where the preferences would lie, it is wonderful to consider how different the campaign would have to have been with choices depending on real information, but even if a majority vote emerged for one of the continuum, ‘not that one’ would always be the biggest bloc vote, and there would always be a continuing rearguard action from the ‘defeated’!

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