The future of the United Kingdom could depend less on the coming Scottish referendum than on the outcome of some apparently arcane negotiations between Edinburgh and London over the next few weeks. In my typically modest fashion, I am calling the problem that needs to be solved, the Kellner conundrum. Here goes.



Much of the news in recent days has reported the spat over the timing of a referendum on Scottish independence. In truth, the far bigger question is whether the Scots will be offered two options or three. David Cameron wants a straight choice: the status quo or full independence. Alex Salmond wants a third option - 'devo-max'. This would keep Scotland inside the United Kingdom, but the Edinburgh parliament would have far greater powers, especially over taxation and public spending.



A simple two-option referendum is obviously straightforward. When the votes are counted, there can be no doubt about which side won and which lost. But this would deny the Scots the chance to vote for a future that many of them would prefer. Salmond is a popular and persuasive politician with a strong case: I would not bet against him achieving a three-option referendum.



This is where the Kellner conundrum kicks in. There are at least four different ways in which a three-option referendum can be held. And the choice of system could decide the outcome.



1. First-past-the-post

Under FPTP, Scots would put a cross against their favourite system, just as they plump for their favourite candidate in general elections. Suppose this produces the following result:



Status quo: 34%



Devo-max: 30%



Independence: 36%



The most popular choice is independence. Scotland would go its own way, even though fewer than 50% of Scots voted for the UK to break up.



2. Alternative vote

This is the system proposed for Westminster and rejected in last May’s UK-wide referendum. People put a '1' against their favourite option/candidate, '2' against their second, and so on. The winner needs 50% support.



Suppose my FPTP example above reflects first choices. None of the options passes the 50% mark, so the least popular choice, devo-max, is eliminated. This would bring into play the second choices of the 30% who put a '1' against devo-max. Suppose they divide 20-10 in favour of the status quo rather than independence. Then the final outcome would be:



Status quo: 34% + 20% = 54%



Independence: 36% + 10% = 46%



On these figures, the status quo would prevail.



3. Two questions

This is the method that Alex Salmond favours. People would be asked two questions: would they favour or oppose a move from the status quo to devo-max; and would they favour or oppose full independence?



I would expect a large majority of supporters of independence to prefer devo-max to the status quo. In my example, suppose the 36% supporting independence divides 28-8% for devo-max. Then the referendum would produce a clear 58-42% majority for devo-max over the status quo, while (assuming the same calculations as in system 2 above) rejecting independence.



However, a strong Salmond campaign could change the numbers: this system gives Salmond the outside hope of a vote for independence.



4. Condorcet voting

This is probably the most democratic system but least likely to be used: I can't see a British Prime Minister agreeing to an unfamiliar idea thought up by an eighteenth century French philosopher.



Think of the contest as a mini soccer league, with each option as different club. Under the Condorcet system, there would be three votes, equivalent to the three matches that would be needed for each option to 'play' each other. Thus voters would be asked three questions: a) would they prefer the status quo or devo-max? b) the status quo or independence? c) devo-max or independence?



In my scenario, these would be the results of the three votes:



Devo-max beats the status quo (assuming most fans of independence prefer devo-max to the status quo)



The status quo beats independence

Devo-max beats independence



So two wins for devo-max and one for the status quo. Devo-max tops the league and collects the victory cup.



This is like Salmond's favourite, system 3, but with one extra vote: devo-max versus independence. I can see why Salmond would prefer to avoid it. He hopes that smart campaigning will push support for independence, as against the status quo, above the 50% mark. But he knows that devo-max is bound to beat independence if the two options go head-to-head (because the vast majority of status quo supporters would switch to devo-max in order to prevent independence).



By avoiding the full Condorcet monty, he avoids a system that would certainly kill his dream of early independence. Proposing a system that pitches independence against the status quo, but not independence against devo-max, is like pitching Salmon’s favourite soccer team, Hearts, against Dunfermline but not Celtic – the easier contest but not the tougher one. You have to hand it to Scotland’s First Minister: sounding utterly reasonable while proposing the particular system that will unquestionably mean more power for Edinburgh, and gives him an outside chance of full independence.

Four Systems - three outcomes

There you have it: four different systems, three different outcomes. And behind the arithmetic lies a genuine argument about the nature of democratic choice when three options are on the table: do we go with the most popular choice, the majority choice that’s left when the least popular has been ruled out, or the option that comes out on top when the contest is run as a pair of alternatives to the status quo?



In my example, independence is the most popular choice; the status quo is the victor when devo-max, the least popular, is eliminated; and devo-max probably wins out when voters are asked two questions - and definitely when they are asked three. There is no obviously right winner: genuine democrats could hold different views about which should prevail.



It would be nice to think that the coming weeks will see an earnest and elevated discussion about democratic principles. Sadly, we all know that Salmond, Cameron and their colleagues will be guided more by likely outcomes rather than democratic theory. Salmond is unquestionably one of the smartest politicians in Britain today. I'm sure he understands - and has understood for years – the arguments I have outlined here. He favours a three-option, two-question referendum because it serves his ambitions best.



Whether Cameron understands this is another matter. If he didn't before, I’m sure that one of his advisers - possibly Andrew Cooper, the shrewd former head of Populus – will brief him before he meets Salmond later this month. Will Cameron continue to resist a three-option referendum - or, as the price of accepting a three-option vote, insist on a voting system less certain to reject the status quo? The 'best' system from Cameron's point of view is AV - but could he embrace the system that he was roundly condemning just a few months ago without looking ridiculous? Cameron’s second best alternative is Condorcet - but could he bring himself to bring French thinking to solve a UK problem?



However they tackle the Kellner conundrum, the underlying point remains: the future of Scotland and the United Kingdom may be decided by the talks over the coming weeks, more than by the referendum campaign itself, whenever it takes place.



Cross-posted with thanks from Peter Kellner's political commentaries at YouGov.