Some people advocate a simple n-way ballot like:
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?
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%.
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.
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.
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.
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.
