Applying voting philosophies to Brexit options

The earlier discussion on Proportional Representation versus District Representation has resulted in this paper: Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats, MPRA 80297.

Brexit stands out as a disaster of the UK First Past The Post (FPTP) system and the illusion that one can use referenda to repair disproportionalities caused by FPTP. This information is missing in the otherwise high quality overview at the BBC.

In the earlier Puzzle on the YouGov poll I estimated Brexit preference orderings from a summary statistic published by YouGov. The next step is to use these orderings for the various voting philosophies. I will be using the website of Rob LeGrand since this makes for easy communication. See his description of the voting philosophies. Robert Loring has a website that referred to LeGrand, and Loring is critical about FPTP too. However, I will use the general framework of my book “Voting theory for democracy” (VTFD), because there are some general principles that many people tend to overlook.

Input format

See the Puzzle weblog text for the problem and the excel sheet with the estimate of the preferences and their weights. LeGrand’s website now requires us to present the data in a particular format. It seems best to transform the percentages into per-millions, since that website seems to require integers and we want some accuracy even though the estimate is tentative. We can also drop the preference rankings with zero weights. Thus we get 14 nonzero weighted options. We enter those and then click on the various schemes. See the YouGov factsheet for the definition of the Brexit options, but for short we have R = Remain, S = Soft / Single Market, T = Tariffs, H = Hard / WTO. Observe that the Remain options are missing, though these are important too.


Philosophy 1. Pareto optimality

The basic situation in voting has a Status Quo. The issue on the table is that we consider alternatives to the Status Quo. Only those options are relevant that are Pareto Improving, i.e. that some advance while none lose. Commonly there are more Pareto options, whence there is a deadlock that Pareto itself cannot resolve, and then majority voting might be used to break the deadlock. Many people tend to forget that majority voting is mainly a deadlock breaking rule. For it would not be acceptable when a majority would plunder a minority. The Pareto condition thus gives the minority veto rights against being plundered. (When voting for a new Parliament then it is generally considered no option to leave the seats empty, whence there would be no status quo. A situation without a status quo tends to be rather exceptional.)

In this case the status quo is that the UK is a member of the EU. The voters for R block a change. The options S, T and H do not compensate the R. Thus the outcome remains R.

This is the fundamental result. The philosophies in the following neglect the status quo and thus should not really be considered.

PM 1. Potentially though, the S, T and H options must be read such that the R will be compensated for their loss.

PM 2. Potentially though, Leavers might reason that the status quo concerns national sovereignty, that the EU breaches upon. The BBC documentary “Europe: ‘Them’ or ‘Us’” remarkably explains that it was Margaret Thatcher who helped abolish the UK veto rights and who accepted EU majority rule, and who ran this through UK Parliament without proper discussion. There seems to be good reason to return to unanimity rule in the EU, yet it is not necessarily a proper method to neglect the rights of R. (And it was Thatcher who encouraged the neoliberal economic policies that many UK voters complain about as if these would come from the EU.)

Philosophy 2. Plurality

On LeGrand’s site we get Plurality as the first step in the Hare method. gets 35% while the other options are divided with each less than 35%. Thus the outcome is R.

(The Brexit referendum question in 2016 was flawed in design e.g. since it hid the underlying disagreements, and collected all dissent into a single Leave, also sandwiching R between various options for Leave.)

Philosophy 3. Hare, or Instant Run-off, a form of Single Transferable Vote (STV)

When we continue with Hare, then R remains strong and it collects votes when S and H drop off (as it is curiously sandwiched between options for Leave). Eventually R gets 44.4% and T gets 55.6%. Observe that this poll was on June 12-13 2017, and that some 25% of the voters “respect” the 2016 referendum outcome that however was flawed in design. I haven’t found information about preference orderings at the time of the referendum.

Philosophy 4. Borda

Borda generates the collective ranking S > T > R > H. This is Case 9 in the original list (including zero weights), and fortunately this is single-peaked.

Philosophy 5. Condorcet (Copeland)

Using Copeland, we find that S is also the Condorcet winner, i.e. wins from each other option in pairwise contests. This means that S is also the Borda Fixed Point winner.


The major point of this discussion is that the status quo consists of the UK membership of the EU. Part of the status quo is that the UK may leave by invoking article 50. However, the internal process that caused the invoking of article 50 leaves much to be desired. Potentially many voters got the suggestion as if they might vote about membership afresh without the need to compensate those who benefit from Remain.

Jonathan Portes suggested in 2016 that the Brexit referendum question was flawed in design because there might be a hidden Condorcet cycle. The YouGov poll didn’t contain questions that allowed to check this, also because much has happened in 2016-2017, including the misplaced “respect” for the outcome of a flawed referendum. A key point is that options for Remain are not included, even though they would be relevant. My impression is that the break-up of the UK would be a serious issue, even though, curiously, many Scots apparently rather prefer the certainty of the closeness to a larger economy of the UK rather than the uncertainties of continued membership of the EU when the UK is Leaving.

It would make sense for the EU to encourage a reconsideration within the UK about what people really want. The Large Hadron Collider is expensive, but comparatively it might be less expensive when the UK switches to PR, splits up its confused parties, and has a new vote for the House of Commons. The UK already has experience with PR namely for the EU Parliament, and it should not be too complex to use this approach also for the nation. Such a change might make it also more acceptable for other EU member states if the UK would Breget. Nigel Farage much benefited from Proportional Representation (PR) in the EU Parliament, and it would be welcome if he would lobby for PR in the UK too.

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 quite maltreated. As confused as the British have been on Brexit, they might also be sensitive to a “stab-in-the-back myth”.


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