Thomas Colignatus has written extensively about the flawed design of the referendum question put to UK voters in June 2016,¹ and in recent months many voters have admitted that they were not clear what they were voting for and may even be having second thoughts.² In this article, Thomas looks at the results of a YouGov poll taken just after this year’s General Election which sought to elicit people’s preferences for various forms of Brexit. Predictably, perhaps, the results show a degree of incoherence and inconsistency. For instance, some 17 per cent of voters ranked ‘remaining’ between different versions of ‘leaving’.

On June 12-13, close to the general election on June 8 2017, YouGov.com (2017) conducted a poll on Great Britain’s preferences on Brexit options. This poll gives an insight in voters’ views at a crucial moment in time. YouGov’s original publication only gave a summary table, and I thank Anthony Wells for making the full preference data available. The data are in the Appendix below.

4 options and 24 preference orderings
YouGov (2017:13-16) gives a clear description of the four options, so that those polled would not need to suffer confusion about their meanings. We use the labels:

R = Remain;
S = European Economic Area (EEA) aka Single Market aka ‘Soft’;
T = Tariffs aka ‘Hard’;
N = No Deal, World Trade Organisation (WTO).

Hence, there is a logical order R, S, T, N, in terms of the political and economic distance between the UK and the EU.

With A > B we denote that A is strictly preferred over B. A consistent Remainer would tend to have the ranking R > S > T > N, and a consistent Leaver would tend to have this in reverse. With 4 options there are 4! = 4 x 3 x 2 x 1 = 24 possible strict preference orderings, excluding indifferences. Preference orderings can also be denoted with rankings (1st, 2nd, 3rd, 4th) and with ordinal utility levels (4, 3, 2, 1). The 1st preferred has the highest preference value 4. We will look with special attention at Cases 1, 5, 9 and 24 mentioned in the Appendix. Only 25 per cent have the preference R > S > T > N and only 15 per cent its reverse, so that 60 per cent has something in between.

Table 1 gives the original results. (For a wider discussion see Wells, 2017). We find that R has dropped as a first preference from 48 per cent at the 2016 referendum to 35 per cent now. These Wells calls the ‘Re-Leavers’. Apparently 26 per cent of those polled express explicitly that they respect the outcome of the referendum and adjust to the current situation of ‘Leave’, which for some may run against the objective of the poll to discover what their first preference still is. 47 per cent put R last, while 17 per cent (a 1 per cent missing due to rounding) (and more than 48 – 35 = 13 per cent) has R sandwiched between options of L. One would wonder how such people would vote in a referendum when they are presented with only two options R or L.

Discerning the collective preference

There are various methods (including elections) to turn the orderings and their weights into a collective preference. What one calls the ‘GB volonté générale’ depends upon what method one selects. Let us consider some major methods, using Rob LeGrand’s (undated) online calculator.

• Pareto Optimality: Everything that is better than the status quo. Since the status quo is EU membership, the 35 per cent for R can block a change, unless they would be compensated for their loss. Part of the status quo is that the UK may leave by invoking article 50. However, the process that caused the invoking of article 50 leaves much to be desired. The 2016 referendum question was flawed, see Colignatus (2017a). For example, voters might have got the impression they might vote about membership afresh without the need, under the Pareto principle, to compensate those who benefit from Remain. The Re-Leavers apparently show loyalty to a flawed referendum question.

• Plurality (the system used in districts for general elections): L is dispersed, and 35 per cent for R is higher than each of the others.

• Instant Run-Off: S and N drop out and eventually R gets 45 per cent and T gets 55 per cent.

• Borda: This selects Case 9, with is S > T > R > N. Only 2.2 per cent actually expressed this directly but Borda selects it as the GB collective preference.

• Condorcet (Copeland): S is also the Condorcet winner, i.e. wins from each other option in pairwise contests.

• Borda Fixed Point: The combination of Borda and Condorcet implies that S is also the Borda Fixed Point winner (excluding considerations on the status quo).

Single peakedness
Figure 1 plots Case 9 or the Borda choice, putting the logical order R, S, T, N on the horizontal axis and the ordinal utility levels on the vertical axis. The connecting lines are only for readability, so that we can better see the plotted dots. The lengths of the connecting lines have no meaning since both axes are ordinal.

Case 9 is curious since it sandwiches R between S, T and N. It still has a single peak that respects the logical order however. Option S would be a ‘bliss point’, with lower utility on both sides. Economic theory has a core belief that voters are rational, so that any preference would have some logic. The logical order R, S, T and N on the horizontal axis might seem arbitrary to some voters who may think otherwise. We do not impose that order but invite voters who think otherwise (these would be at least 8.8 per cent) to explain why they choose a different order. Potentially each voter has his or her own criteria to create logic and order. However, voters with multiple peaks on the common logical order would have more to explain to us, merely for us to understand them, than voters with a single peak. Without a good explanation, we cannot reject the possibility that there is some confusion. For example, it is not clear what to think about R > N > S > T (Case 5, 1.3 per cent). This would be Remainers who would rather prefer No Deal to the EEA or some agreement not to have a trade war on tariffs. A tentative explanation is that these voters have a somewhat binary position, as Remain versus No Deal At All, while the other options are neglected.

Summary overview, spotting likely confusion
Table 2 categorises the preference orderings on two rational perspectives, with a remainder that subsequently is identified as confusing on both rational perspectives. The columns distinguish orderings with R first and R last, which would seem to be rational, and thus a remainder (17.3 per cent) with a sandwich that raises eyebrows. The rows distinguish between: (i) cases 1 and 24, that are the rational opposites, (ii) the other single peaked cases, that thus respect the logical order of the horizontal axis, and (iii) the remaining multiple peaks (26.8 per cent), that thus presume some unknown logic. We find 8.8 per cent with a sandwich with multiple peaks, that defy understanding on both counts.

Potentially, the 8.8 per cent confused, as I tend to identify them, are only present amongst the Re-Leavers or the non-voters. However, the same pattern can be seen amongst 2017 voters for the Conservative Party and Labour, who also took part in the 2016 referendum (ConR/L, LabR/L). In 2017 all 8.8 per cent are L, and in 2016 at least 50 per cent would be L at the time of the referendum, notably when we focus on ConL and LabL. Potentially the outcome of the 2016 Brexit referendum has been decided by this group. This 8.8 per cent concerns 145 people in a poll of 1651 GB adults, while perhaps 4 per cent uncertainty as a rough rule of thumb might be wise to assume for polls in general (unless you happen to hit upon the right poll). I am not aware of the existence of such a poll at the time of the referendum, and we really do not know for sure.

A perspective from Proportional Representation (PR)
Clearly this exercise hinges strongly upon the uncertainties in polling, in which those polled did not answer the question with the same amount of rational attention as we are trying to discern here. Yet the exercise draws attention to aspects that would benefit from attention by policy makers.

The UK has District Representation (DR) and Holland has Proportional Representation (PR). I tend to regard Brexit as an example of how DR tends to cause mishaps. In 2015, UKIP got 12.6 per cent of the vote and only 1 seat. UKIP started to threaten the seats of the Conservative Party, and David Cameron, no supporter of Brexit, called a referendum in the hope that this would put the issue to rest. It might have done so, but the referendum question was flawed. We still have no idea how the UK electorate thinks about the options for Remain. A referendum gives proportions but referenda tend to be silly and dangerous, as they are an instrument of populism rather than of representative democracy. The June 8 2017 general election was again a severe mishap. The PR Gini for the UK was 29.7 per cent in 2015 and this improved to 15.6 per cent in 2017, since voters returned to the model of voting either Labour or Conservative. Yet this also involves strategic voting at the cost of first preferences. This compares to the Dutch 3.6 per cent in 2017, see Colignatus (2017b). The UK might be advised to adopt PR (also used in EU elections) and let a House of Commons elected under PR reconsider Brexit. The other EU Member States might accept a Bregret when the UK shows what major change was required to resolve the confusion (of only 8.8 per cent?).

Nevertheless, given the observable tendency in the UK to prefer a soft Brexit, the EU would likely be advised to agree with such an outcome, or face a future with a UK that rightly or wrongly feels maltreated. As confused as many Brits have been on Brexit, they might also be sensitive to a ‘stab-in-the-back myth’.

Appendix: YouGov(216:13-16), full data

Also available as https://boycottholland.files.wordpress.com/2017/09/2017-09-18-yougov-rankings-full-data.xls.

Code letter = Code words = YouGov’s text actually used in the poll:

R = Remain = ‘Britain remaining a full member of the European Union’

S = EEA / Soft = ‘Britain leaving the EU but remaining in the single market, giving free trade on our exports in exchange for following some rules, paying a fee and allowing immigration from the EU’

T = Tariffs / Hard = ‘Britain leaving the EU and having only a limited trade deal, giving Britain control over our borders, but putting barriers or tariffs on some of our exports’

N = No Deal = ‘Britain leaving the EU and not having any special trade deal, giving Britain complete control over our rules and borders, but facing barriers or tariffs on our exports’

The 1651 polled also ccontain non-voters. The labels ConR/L and LabR/L are for those who voted Conservative or Labour in 2017 and who voted R or L in 2016.

Notes:

1. Including, in these pages, RES Newsletter no. 177, April 2017. Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland. This article summarises Colignatus (2017c, d) in the references. The editor and author are grateful to Carlo Perroni at the University of Warwick for comments on an earlier draft of this article.

2. For example, on 1st September 2017, the Financial Times reported Sue Shaw, a school worker in Staines, as saying that she had expected Brexit by last Christmas. ‘“A lot of people I work with thought we would be out within a week, they thought that if we voted leave we would get on and leave,” she said.’

References:

Colignatus, Th. (2017a), “Voting theory and the Brexit referendum question”, RES Newsletter, Issue 177, April, p14-15

Colignatus, Th. (2017b), ‘Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats’, https://mpra.ub.uni-muenchen.de/80297/

Colignatus, Th. (2017c), ‘YouGov Brexit ranking data of June 12-13 2017’, https://boycottholland.wordpress.com/2017/09/18/yougov-brexit-ranking-data-of-june-12-13-2017/

Colignatus, Th. (2017d), ‘Applying voting philosophies to Brexit data’, https://boycottholland.wordpress.com/2017/09/19/applying-voting-philosophies-to-brexit-data/

LeGrand, R. (undated), ‘Ranked-ballot voting calculator’, http://www1.cse.wustl.edu/~legrand/rbvote/calc.html

YouGov.com (2017), ‘YouGov survey results 170613’, https://d25d2506sfb94s.cloudfront.net/cumulus_uploads/document/9pum7c5c4j/AnthonyResults_170613_Brexit_W.pdf

Wells, A. (2017), ‘Public opinion on Brexit’, UK Polling Report, June 18, http://ukpollingreport.co.uk/blog/archives/9921