Some people advocate a simple n-way ballot like:

  • Remain
  • Deal
  • No deal

But this splits the leave vote.

An alternative is to use Single Transferable Vote, which in the case of a referendum would be an Instant Runoff/Alternative Vote election if I understand correctly.

Would this voting system also be open to criticism of vote-splitting?

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    I don't think that "Would this specific election be subject to vote splitting?" is an exact duplicate of "What arguments are there against ranked-choice voting?" In particular, we could discuss how polling results suggest that people would vote in IRV or other systems. – Brythan Oct 23 '18 at 1:52
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    Well a Two-round System could also be considered. The first vote would eliminate all but the most voted two options. The second vote would chose the winner. A lot of countries use this system to elect Presidents but I've never seen it used for a referendum. – armatita Oct 24 '18 at 9:31
  • The problem I see with every nominated mechanism is that if “Remain” is on the ballot, Leavers end up in a Hobson’s Choice situation, whereby they will likely be compelled to vote for something they don’t want, whereas the Remainers have the precise option they want. Importantly, there is similar variance in the possible outcomes for Leave and Remain. In other words, “Remain” covers a multitude of futures, just as Leave does, but having a ballot option of “Remain” leave Remain futures open, but closes Leave futures. Hence a perceived unfairness. – 52d6c6af Oct 24 '18 at 11:04
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    @Ben the maximum number of options would likely be four (No deal, Canada, Norway, EU member). I don't think an EU member vote has ambiguity (you are either a member, or not). And yes, 3 leave options would split the vote. That's why I believe a two-round referendum would be appropriate. The two most voted options pass to the second stage. At this point the voters will need to chose either between two Leave scenarios, or between one Leave and Remain. This is not just about remaining or leaving, but also in what terms. (note: voted to reopen). – armatita Oct 24 '18 at 13:23
  • Should we interpret "simple n-way ballot" as first past the post? Approval voting is also fairly simple for example. – origimbo Oct 24 '18 at 14:23

It is not possible to have a truly fair vote between three or more alternatives.

Arrow's Impossibility Theorem states:

when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.

Let me unpack that.

  • Ranked voting electoral system: each voter can put the options in order of preference. Given options X, Y and Z one voter might select Y first, X second and Z third, while another voter might prefer X first, Y second and Z third. A voting system where you simply chose your top preference is an example of this, as are things like Single Transferable Vote and run-off systems. Voting systems that are not ranked are points systems, where for instance you are asked to allocate 100 points between the different options.

  • Community-wide ranking: the voting system must put the available options in order: first place, second place, and so on. "Complete" means that the results must include all the options, and "transitive" means that if X beats Y and Y beats Z then X must also have beaten Z. These things sound obvious, but mathematicians like to be precise about them.

  • Unrestricted domain: The voting system has to come up with an answer no matter how people vote. Its not allowed to say that there is no winner (although it is allowed to declare a tie if vote counts are exactly equal).

  • Non-dictatorship: The voting system isn't allowed to be just a single voter. Again, its one of those obvious things that mathematicians feel the need to write down.

  • Pareto efficiency: if everyone prefers X to Y then X should beat Y.

  • Independence of irrelevant alternatives: Suppose we are voting between X and Y, and X beats Y. Now we run the election again with a third candidate Z. Each voter still puts X and Y in the same order as the first time, but they also add Z to their list of preferences somewhere. If we do this then X should still beat Y, no matter where Z comes.

The theorem states that it is impossible to come up with a voting system that meets all these criteria.

In the question there are three alternatives offered: "Remain", "Deal" and "No Deal". Arrow's Impossibility Theorem says that there is no voting system which can produce the "right" answer given all possible combinations of votes. For instance, you might find that we get "No deal" even though everyone would prefer one of the others.

This also means that the voting system choice is a loaded question. The people in charge of running the vote can ask a selection of voters how they would vote on the three options and then pick a voting system which, given those preferences, produces their preferred answer.

For instance suppose:

  • 45% of the population would vote for 1: No Deal, 2: Deal, 3: Remain.

  • 35% would vote for 1: Deal, 2: Remain, 3: No Deal.

  • 20% would vote for 1: Remain, 2: Deal, 3: No Deal.

Now which do you want to win?

  • No Deal: use First Past The Post. No Deal wins 40% of the vote, with the other two getting 35% and 20%.

  • Deal: use Instant Runoff: After the first round "Remain" gets eliminated and its 20% of the votes get added "Deal", which now wins with 55%.

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  • +1 This is an excellent answer, and unpacking of Arrow's Impossibility Theorem, but I think framing it as "It's impossible to have a fair system" is a little misleading. While it's certainly impossible to have a perfect system, pretty much every country has settled on a system it sees as "fair enough" for it's elections, so using such a system is in no way an impossibility. That said I don't want to come off as too critical cause this really is a great answer. – CoedRhyfelwr Oct 24 '18 at 16:38
  • Thank you. Honest question: how do you know this? Do you have training in this area? Also: if the system of voting is loaded, can the options presented in the vote also be loaded, meaning that both the system and the options must be carefully chosen to avoid accusations of unfairness? – 52d6c6af Oct 24 '18 at 17:04
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    @Ben "How do I know this?" Well, many years ago a colleague told me about it, and since then I've read the Wikipedia article I referenced, plus one or two other things. Can options be loaded? Yes, of course. "Do you want to damage this country by leaving the EU?" is a loaded question. So is "Would you like to free up £350M/week for the NHS by leaving the EU?" – Paul Johnson Oct 24 '18 at 21:31
  • What about approval voting? – hkBst Nov 4 '18 at 18:24
  • @hkBst According to Wikipedia it only manages pareto efficiency under the strong Nash equilibrium condition (perfect information, perfectly rational voters), which strikes me as unrealistic. So probably not. – Paul Johnson Nov 4 '18 at 22:22

Let us consider potential voting systems:

Simple Plurality

This is the system initially described. Simply put, the option with the most votes wins. Using this system would be a terrible idea since it would split the leave vote multiple ways, and you would very possibly end up with a decision that a minority of people voted for. It would trigger cries of betrayal from leavers immediately, and the result would likely not be respected. This will not happen.

Instant-Runoff Voting

This avoids vote splitting. Everybody would be able to rank the options. If there is no majority, the least liked option on first preferences would be eliminated, and those ballots' second preferences would be used. Repeat until a majority is found. This avoids vote splitting, and only requires one election. There are some issues with this voting method, including some bizarre tactical voting possibilities (see the wiki article on it here) but generally, it seems the fairest possible option for this. One of its main advantages is its simplicity. Other (potentially better) systems are more complex to compute, and so are more open to allegations of corruption as the general electorate may not be able to easily work out how the result was reached.

Two-Round Runoff Election

In this version there are two election days. On the first, all options are available. Then the top two are put head to head on the second. This is essentially a more expensive, less representative version of IRV. It is, however, even simpler, and it favours options which do very well on first preferences. Some people see that as more representative of people's true wishes, and it is more familiar to voters used to plurality systems. In a 3 option referendum, it would play out very similarly to IRV. The only legitimate difference would be the added expense of two separate election days.

While there are many other voting systems, most require more complex assessment to determine the victor. This would lead to questions of the legitimacy of the result, since people like to be able to immediately tell how their vote was counted. While systems such as Schulze or Borda Count may provide more representative results, perceived opacity in their computation is such that in my opinion, their use is highly unlikely in such a circumstance.

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  • Thank you. It seems to me that having a single option for Remain versus many options for Leave is dishonest because it implies that there is more consensus around possible futures for Remain than there is for futures for Leave. Can you see my point? Put another way: does the fairness of an election depend also on the options on the ballot (in addition to the system used). – 52d6c6af Oct 24 '18 at 15:30
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    I can see your point, but I respectfully disagree - while the future may be uncertain, it is not related to the action that must be taken now. The fundamental question of the vote would surely be "What kind of a deal would we want?" In that case, remain is the natural complement to no deal. If thought about in terms of a rerun of the original referendum, it does seem unfair, but if treated as it's own question, with it's own answers, I think it makes a lot of sense. That could just be my bias though! I personally can't think of multiple, sensible remain options, but maybe others can. – CoedRhyfelwr Oct 24 '18 at 15:35
  • I can quickly think of different Remain options: 1. Remain, but no more integration 2. Remain, but continued integration 3. Remain, with specific limits on further integration – 52d6c6af Oct 24 '18 at 17:02
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    @Ben the issue with your possible remain options is exactly the same on the government is currently struggling with over the terms of the leave deal. The choice between them is not something the UK can decide for itself. 28 nations have to make the choices about how they want to move together. No one country can dictate that direction. Even f we did go with your outline, number 3 is basically what caused this mess in the first place as specific limits could never be defined or agreed prior to a referendum, just like what leave actually meant was never defined. – Jontia Oct 24 '18 at 21:58
  • What about approval voting? – hkBst Nov 4 '18 at 18:24

I'm going to answer this by rejecting the premise of the question, which is that splitting the "Leave" vote into deal or no-deal gives an unfair advantage to remain and is therefore unacceptable.

Vote splitting is only relevant if the distinction between the two positions is minor and/or artificial on one side of the issue, while divisions on the other side of the vote have been ignored or removed to artificially collect separate groups of voters together.

The vast difference between leaving the EU with a deal that may include regulatory alignment, customs unions, freedom of movement etc and leaving on no deal means this split is ideological not artificial and should be represented in a vote. If anything the original referendum could be criticised for clumping so many variations and degrees of change under the simplistic "leave" heading.

There is absolutely no need to be concerned about vote splitting between two distinct positions which "Deal" and "No Deal" represent given the very different consequences each choice would have on Britain and indeed the EU.

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    Doesn't it depend on the contents of the deal? A deal could range from BrINO (brexit in name only, all stays the same except for some things only EU member states have, e.g. voting in the EP) to extremely hard brexit with some small cooperation agreement (e.g. like the EU-Russia PCA). Depending on that, two options as listed in the question could be seen to be (very) similar. – JJ for Transparency and Monica Oct 24 '18 at 14:10
  • @JJJ To some extent yes. Pragmatically a "Deal" position is just as likely to split the Remain vote depending on content as it is a "Leave" vote. Which is why it is fine to have the actual options on the table if you have to have a referendum in the first place. Which is almost never a good way of taking complex decisions. – Jontia Oct 24 '18 at 14:12

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