Mathematician Bernhard von Stengel (LSE) uses game theory to consider how a second Brexit referendum with more than two choices could be run, and how the counting-rule chosen for any multiple-choice ballot can determine the outcome.
If you favour a second referendum on Brexit (a prospect that is now, in early 2019, receding), you should not only think of what you should ask the people, but how you would reconcile their choices. This is a central question of Mathematical Social Choice, with it attempts to answer since the middle ages. The question becomes interesting when there are more than two choices on the ballot paper. Suppose the choices are:
- D = leave the EU with a negotiated Deal (also called ‘soft Brexit’),
- N = leave the EU with No Deal (a ‘hard Brexit’), or
- R = Remain in the EU.
Every voter is given a first and second choice which represents their most and second-most preferred outcome. An example of a ballot is here:
On this ballot, the voter has expressed their choice as 1. R (Remain) and 2. D (Deal) with No Deal as the implicit third choice.
Assume we have 9 voters (or equal-sized voting groups) who have the following preferences:
Here, the first 4 columns are Remain voters whose preference is 1. Remain, 2. Deal, 3. No Deal. The next 3 columns are ‘Hard Brexiters’ whose preference is exactly the reverse. The last 2 columns are voters whose first preference is Deal, with one of them having Remain as their second choice, the other No Deal as second choice. These preferences and their distribution are not unrealistic. A voting rule now tells us how to distil ‘the will of the people’ from these preferences. But which rule should we choose? There are several contenders for such a rule.
Plurality, also called ‘first-past-the-post’
This declares as winner the option that has gotten the most first-choice votes (so one does not even need a second choice and the ballot paper is simpler). In parliamentary elections in the UK, the MP representing a constituency is chosen in this way. Here the most first-choice votes (four out of nine) are for Remain, but this is not a majority of all votes – five out of nine would rather leave the EU without or with a deal.
Supplementary vote, also called ‘Instant Runoff’
This means that the option that gets the least first votes is discarded, and the second preference of the voters who made that choice is counted (as if they would be asked to vote again in a ‘runoff’ vote, assuming that the others stay with their first choice). Here, these are the voters who chose ‘D’ and their votes are split, one of them for ‘R’ and the second for ‘N’. The total is now five for Remain and four for No Deal, and Remain is the winner as the declared ‘will of the people’ according to this rule.
The rule seems clear and fair enough, but it has its problems. The main problem is called strategic voting which means that voters have an incentive to misstate their true preference. Namely, the minority (of people who chose ‘D’) now have in effect the casting choice between two polarised outcomes. If the above preferences were known (supported by opinion polls, say), then an ‘N’ voter as above would have an incentive to misstate their preference instead as 1=D 2=N 3=R (that is, swap their first and second choices), to let N become the decisive minority with 2 out of 9, after D which now has three out of nine first choices. The other 2 N voters would both choose D and create the final vote 4 R versus 5 D, meaning to the leave the EU with a deal. While it remains doubtful that voters are that strategic, it would, on the other hand, create an incentive to be a bit more moderate. However, not all voting rules favour R for the above voter preferences.
Condorcet winner, or pairwise comparison
This rule looks at the stated order of preferences and compares any two outcomes with each other. That is, the preferences of the voters are now used to answer a question such as ‘do you prefer D over N’?
Here we get the following answers:
- D beats R by 5 against 4 votes
- D beats N by 6 against 3 votes
- R beats N by 5 against 4 votes
This gives the following clear collective preference: a strict majority prefers D over R, and another strict majority prefers R over N, and another strict majority prefers D over N (which does not follow from the first two, see below). Here the ‘will of the people’ is D first, R second, N third. Sounds great, but D was a first choice for only two out of nine voters. Does this rule head for mediocre choices? Or for useful compromise?
The Condorcet rule has something less desirable than any of the other rules: It may not produce a clear winner but create cycles, as the following modified voter profile shows (three voters suffice):
Here, two-thirds of voters prefer R over D, two-thirds prefer D over N, and two-thirds prefer N over R. Such voter preferences may not be realistic, but who knows? The fact that they are theoretically unavoidable for any reasonable voting system is known as ‘Arrow’s Impossibility Theorem’, after the economist Kenneth Arrow (1921–2017).
Borda count, or giving points
Yet another voting system tries to avoid cycles by giving points for first and second choice, for example two points for first choice, one point for second choice, zero for third choice (instead of points 2,1,0 we could also give 3,2,1 with the same effect, which is just an extra point everywhere). The option with highest total number of points wins.
In our 9-voter example, this gives points
- R = 2+2+2+2+0+0+0+1+0 = 9
- N = 0+0+0+0+2+2+2+0+1 = 7
- D = 1+1+1+1+1+1+1+2+2 = 11
which makes again D the winner. But hey, what if someone does not put an ‘X’ for their second choice at all? How should that shift the points? You get only one point for your first choice, and zero for the others? Or two points for your first choice, and zero for the others (which would surely let the Remain voters above drop their points for ‘D’ in second place).
Or we could, like in football, give 3 points for first choice, 1 point for second choice, resulting in
- R = 3+3+3+3+0+0+0+1+0 = 13
- N = 0+0+0+0+3+3+3+0+1 = 10
- D = 1+1+1+1+1+1+1+3+3 = 13
with R and D tied. But why this rule?
Some history
The Condorcet method is named after the Marquis de Condorcet (1743–1794), who died in a prison cell, poisoned, during the French Revolution. However, it was already invented in 1299 by the Majorcan polymath Ramon Llull (ca. 1232–1316). To remember that name (which has four letters ‘L’ in it and one vowel) think of the binary number 11011. Llull indeed invented the binary system and is considered by some as the inventor of information theory. He was so enthralled by it that he thought St Mary should be added to the Holy Trinity to make their number a power of two. Heretic stuff that did not make him popular with the church authorities. With the discovery in 2001 of his lost manuscripts, Ars notandi, Ars eleccionis, and Alia ars eleccionis, Llull is given credit for discovering the Borda count (re-discovered several times in later centuries) and the Condorcet criterion.
Conclusion
One conclusion is that you probably shouldn’t put more than two options on a ballot paper, or maybe not hold a referendum in the first place. On what voting system should you agree even to determine the ‘will of the people’, when we have enough trouble to determine it when they made one out of two choices? At any rate, you will appreciate ‘Strong Arrow’s Theorem’ from the geeky cartoon XKCD (one of my favourites):
The post was first published on LSE Department of Mathematics Research blog. It gives the views of the author, not the position of LSE Brexit or the London School of Economics. Image by CGP Grey,(CC BY 2.0).
Bernhard von Stengel is a Professor of Mathematics at the London School of Economics. He teaches discrete mathematics, optimisation and game theory at LSE. He tweets as @bvonstengel.
Some controversial questions are “linear”. By this I mean that you can place everyone on a line. If someone is given a choice between two outcomes, both on the same side of them on that line, they will prefer an outcome closer to them.
For example I think the question of under what circumstances (if any) abortion should be legalised is linear in that way. Someone who wants abortion completely banned would prefer a ban from 12 weeks into pregnancy to one after 24 weeks, and someone who wants abortion completely legal the other way around. (OK, so there will be occasional exceptions. No mathematical rule applies 100% in politics.)
In linear questions a lot of the problems with voting disappear. For example the Condorcet method will give preference to the median position (the position where half the electorate are to one side, half to the other), because at least half the electorate will prefer the median to any other position.
If it hadn’t been for the last two years, one might think that Brexit would also be linear. At one end of the line you would have continuing membership of the EU, on the other complete exit with no deal. Unfortunately it hasn’t turned out that way. Although in some sense the withdrawal agreement lies between the two extremes, a lot of MPs are behaving as it is the worst of all three options. Some have said so publicly, like Dominic Raab. ( https://www.theguardian.com/politics/2018/nov/23/dominic-raab-theresa-mays-deal-worse-than-staying-in-eu ). I see no reason to believe that the general public would behave differently in any referendum.
If there is a moral to this mathematical discourse, perhaps it is that we should try to find a way to linearise the decision space available. But don’t ask me how.
Actually “one-dimensional” would have been a better word than “linear”.
A very good roundup of the types of voting that could be enacted in another referendum. Problem is that it doesn’t address the problem. It would be good for the country to rally around a compromise position where remainders and leavers each get some of what they want but give up some other things. The Noway deal is an example of a compromise – out of the political apects of the EU but in the single market and with free movement.
Theresa May’s deal is unpopular, not because it is a compromise, but because it isn’t a compromise. Hard brexiters don’t like it because there is the threat of the Irish backstop keeping the UK in a kind of quasi semi customs union with no say. Remainers/soft brexiters don’t mind the Irish backstop but are against the deal because the political declaration is for a hard brexit.
There is no evidence that a majority of voters voted to be out of the single market for example. The vote was only for an undefined exit from the EU. If only 10% of leave voters actually would like to stay in the single market but out of the EU, then a majority f the referendum voters would have voted for the single market.
If there is another referendum, it cannot fairly be with Theresa May’s deal as the compromise position as it is not a compromise. If leave with no deal is on the ballot paper, then a real compromise position has to be there as well – so 4 choices.
For my money, if there is a rerun it either has to be remain vs Theresa May’s deal or Theresa May’s deal vs a true compromise (possibly Norway or other). A no deal brexit cannot be contemplated in another referendum. The electorate cannot be allowed a free vote on breaking international treaties such a path only leads to tyranny,
“The electorate cannot be allowed a free vote on breaking international treaties such a path only leads to tyranny,” Which international treaty would a no-deal exit break?
If for example this is the Good Friday Agreement I would appreciate chapter and verse (or a URL) to the relevant passage in the agreement. This is because I have heard (but not verified) that the allegation that the GFA requires no customs controls on the border is a myth. Indeed I find it hard to believe that the signatories to the GFA intended thereby to commit the UK never to exit the EU without the EU’s consent via a deal.
@David. Strange how some people continue to ignore the referendum result. This time it is an mis-interpretation only slightly different from the regular mis-interpretation. Check what was voted on. Cameron, May and Parliament honoured the result, but May and most of the HoC, with support from the Remainers in the Lords, reneged on the clear undertaking to take the UK out of the EU lock,stock and barrel.
The plan was to take back control. Brexiteers voted for the UK to become sovereign and independent. The GFA and the so-called Irish backstop were and are fudges. If, for example, someone makes a choice between becoming self-employed and starting a shop, which he or she had done before for a long time, and remaining employed with a company, and that person decided to go back on their own and start their own business, they do not ask their employer for permission to start a business or change their mind about being self-employed after resigning when a small matter, blown out of proportion by the employer who doesn’t want to let this employee go, is mooted as a problem. Independent and sovereign nation-states deal with issues all the time. That is why they negotiate with other independent and sovereign nation-states about these issues.
Other than humouring Treasa May and a spineless HoC, there is no reason to postpone the exit. The WA deal is not a deal, anyway. It’s a withdrawal agreement which, inter alia, has not been accepted by the HoC.
@jacob jonker. Same old, same old. Take back control, people voted, politicians promised, GFA fudge, example of an individual taking back control of their life.
The world is not so easily categorised, politicians cannot be believed if they promise what they cannot deliver, there is no such thing as taking back control, the vote meant 17.4 millions different things to 17.4 million people who voted leave and analogies are not truth and in your case a vast oversimplification of the problem. We live in the real world, not the world we think we see. Brexit is really a flat earth type of belief.
One thing that no one seems to understand is that actions have consequences, and often those consequences are unintended and exactly the opposite to what was intended. Brexit is the perfect example. In any hard Brexit, especially a no deal scenario the consequences will be having less control, a wholesale destruction of businesses, especially manufacturing businesses, and a dramatic reduction of our wealth – exactly the opposite of what was promised to and expected by leave voters.
@Alias. The treaties I refer to that would be broken by crashing out in a no deal scenario because it was the chosen path and not by accident are the treaties with the EU. The Irish problem, and specifically the GFA, would be a matter of reckless abandonment and a definite show of bad faith, but not perhaps the breaking of the exact letter of the treaty although there are many legal opinions that it would be.
@David: “The treaties I refer to that would be broken by crashing out in a no deal scenario because it was the chosen path and not by accident are the treaties with the EU.” So let me summarise your position. Leaving the EU without the EU’s permission via a withdrawal agreement, or asking the electorate whether they want to do that, would lead to tyranny. Is that right?
I think you have got it the wrong way round. If the UK really were locked into the EU, whatever its citizens wanted, THAT would be tyranny, with the tyrant being the EU. As I have said many times on this blog, this would be undemocratic in the Tony Benn Five Questions sense, in that answer to the question “How do we get rid of you” addressed to the EU would be “you can’t”.
“The Irish problem, and specifically the GFA, would be a matter of reckless abandonment and a definite show of bad faith, but not perhaps the breaking of the exact letter of the treaty although there are many legal opinions that it would be.” Can you cite some of those legal opinions? I found this page from the BBC https://www.bbc.com/news/uk-northern-ireland-46988529 which seems to me fairly good evidence that the GFA does not oblige Eire and the UK to keep their customs arrangements in alignment in perpetuity.