tl;dr version: Agreement requires uncertain cross-party agreement and very good whipping

Why so glum, chum?

At present, betting exchanges imply that the probability of the UK exiting the EU without a deal is around 24%.

That’s about the same as the probability of the average Bundesliga game ending in a draw.

That is already a very high figure for what many people would regard as a very bad outcome.

I worry because I think that figure of 24% is too low. I think the figure is closer to 40%; colleagues whose opinions I greatly respect place it higher at 60–70%.

These numbers are subjective judgements of probability, rather than inferences from past data. Brexit is either unique or sufficiently distinct from other processes as to defeat comparison. There is no other regional trade association which is comparable in depth to the present-day EU, and so we cannot generalize on the basis of past examples of regional disintegration. At the same time, the EU is not a state, and so it is difficult to generalize on the basis of past peaceful secessions (Singapore and Malaysia; Norway and Sweden; Iceland and Denmark).

I think that subjective judgements of probability are worthwhile. Subjective judgements are fallible, and they can convey spurious precision, but these flaws are not unique to subjective judgements, and apply just as well to model-based predictions. Where model-based predictions are not possible, subjective judgements of probability can guide action, and in particular action to insure against negative outcomes.

Subjective judgements do, however, have to be justified. Why do I think the probability of a no deal Brexit is greater than is commonly thought, and why do I think it is lower than others think? I give nine reasons below.

[For clarity: when I talk about a no-deal Brexit, I mean the failure of the UK to approve a negotiated withdrawal agreement with the EU, and/or the failure of the European Parliament to ratify that agreement. I’m purely concerned with agreement at the international level. Although Parliament will need to pass the Withdrawal Bill, I am ignoring that slight complication. A deal which is concluded at international level may still be messy if that deal has not been written into UK law]

First, no deal is the default outcome ("by automatic operation of the law", etc., etc.,). Unless Parliament passes an agreement, and unless the EP ratifies that agreement, the UK will cease to be a member on the 29th March — unless of course an extension is asked for and granted, at which the same logic would repeat at some later date.

Second, a past attempt to reach agreement with the EU failed, and failed badly. When the government brought the current Withdrawal Agreement (WA) to the Commons, it suffered the largest parliamentary defeat of any modern British government. Although a majority of Conservative MPs supported the WA and Political Declaration (PD), a third of Conservative MPs did not, and nor did any of the opposition parties. The scale of this defeat suggests that, whether for policy reasons or for other reasons, we are very far from an agreement which is acceptable to a majority in the Commons.

Third, the EU is unlikely to change the WA, and is only likely to agree changes to the accompanying political declaration; but changes to the political declaration may not persuade MPs because they rightly view the political declaration as subject to further negotiation.

The EU is unlikely to change the WA because it has repeatedly said that it won’t, and because EU positions, once reached, tend to be hard to unpick given the possibility for countries to veto departures from agreed positions. This is true even where the decision-making rule is two-thirds rather than unanimity.

Even if the EU did change the WA (which must be rated as a less than one-in-twenty chance), these changes would, for the same reasons I have just mentioned, be very minor, and would not therefore give many MPs reasons to vote differently.

Fourth, since the WA is unlikely to change, parliamentary approval of a WA requires the cooperation of opposition MPs. Combined, the Conservatives and the DUP have 327 votes, or seven more than they need to pass a motion if all other MPs vote against. But there are more than seven Conservative MPs who would vote down any agreement remotely similar to the present agreement. Any attempt to approve a withdrawal agreement on the basis of Conservative and DUP votes along would be doomed to failure.

Fifth, since the opposition party which is most favourable to Brexit is the Labour party, the most likely route to parliamentary approval runs through Jeremy Corbyn’s party. Corbyn’s past statements on the EU have ranged from tepid endorsement to histrionic opposition, but on this issue I think, along with Rafael Behr, that his primary concern is to avoid being blamed for any negative consequences which might result from Brexit, in whatever form (though party management is a plausible secondary concern). Avoiding blame and managing the party are easier to do if the party is whipped to abstain rather than vote for an agreement. Corbyn's letter is an important step in this direction.

Yet this still means that a deal requires the leaders of the two largest parties to reach an agreement on the most important issue in British politics within the next fifty days, in the knowledge that their agreement is likely only to concern the Political Declaration, an agreement which can therefore be unpicked in subsequent final status negotiations with the EU. The UK generally is not conducive to cross-party agreements; it is easy to see an agreement collapse for lack of trust, or lack of time.

Sixth, even if Labour agrees to abstain (“for the good of the country and to move on to the next stage”), the parliamentary arithmetic remains tight. Had all Labour MPs who voted against the Withdrawal Agreement abstained instead, then the government would have won by 202 votes to 184, a majority of eighteen.

Votes on the Withdrawal Agreement and Political Declaration as they were (left) and as they might be (right)?

But it is unlikely that Labour would be able to ensure that 100% of its MPs abstain; and changes which make Labour more likely to abstain might also make Conservative MPs less likely to vote in favour.

Seventh, even if Labour agrees to abstain, and the parliamentary arithmetic looks right, then there is the possibility that no vote will pass because of poor counting. Estimates of the number of Conservative MPs who will vote against the government on everything except its continuance in office have generally been too low. In an article of October 2018 (“Why Theresa May shouldn’t rely on Labour rebels backing her Brexit deal”), Stephen Bush quoted Steve Baker suggesting that 40 Conservative MPs might vote against the government. Three months later, under more time pressure, the number of Conservative MPs voting against the government was almost three times greater. Conversely, the number of Labour rebels minded to support the Withdrawal Agreement was much less than the “almost thirty” discussed in that same article.

Although the quotes in that article were from October, much later estimates of the scale of the government defeat were also too optimistic concerning the government’s prospects. On average, entrants in John Rentoul's meaningful vote sweepstake said the government would win 39 votes more than it did.

Eighth, if no vote passes because of poor counting, the government may have run out of time to change things. The downside of a strategy of running down the clock is that you have to execute perfectly right before the buzzer sounds. This government may be able to do that — but then again it may not. It would be tragic if the government scheduled a vote on the 27th March confident of its numbers only to fail for the want of a couple of votes.

Ninth, if the government fails to secure parliamentary approval before the 29th March, I do not think that the EU would grant an extension to the Article 50 process. There is some chance that all EU27 MS would agree that Theresa May is close to achieving a parliamentary majority, but this would require that the failure be a very narrow failure, and the path to remedying that failure to be a very clear path, capable of being traversed before the European Parliament sits in plenary on the 2nd July.

These are all ordinary language arguments which could justify a range of subjective probabilities. How can we turn this into something more precise?

One option is to describe the Brexit process as a sequence of decisions, each with probabilities of their own. By listing these decisions, and attaching probabilities to them, overall judgements of probability become better justified. Jon Worth and Jack Lambert have produced flowcharts of this kind.

I don't think it is possible to exhaustively enumerate all the decisions in the Brexit process, for two main reasons.

First, some of the decisions are continuous rather than discrete ("how far will Theresa May accommodate Labour demands?"). That's hard to represent in a flow-chart.

Second, some of the sequences can be played multiple times. There may be a second vote of some kind on the 14th February, or it may be pulled from the parliamentary agenda. If there is a vote on that day, I think it will fail, and so the meaningful "meaningful vote" will be the third such vote — but I can't rule out a fourth such vote. Whilst I can't rule out a fourth vote, I can conclusively rule out writing any flow-chart which features that level of recursion.

Instead, I think it is more helpful to concentrate on the likelihood of what I have described as the most plausible route to agreement, namely co-operation between Conservative and Labour. If that probability is low, then adding on other less likely routes to agreement is unlikely to raise the probability that much.

Let's say that in ordinary English we'd be happy saying that there is a "high chance" that the Conservatives and Labour will reach an agreement about the agreement, and a "high chance" that the vote will be whipped successfully. If a high chance is a probability of around 80% (and some research suggests that's a fair interpretation), and if these two events are independent, then the chance of a deal being reached through this route is .8 * .8 = 64%. If we are prepared to ignore revocation of Article 50, then the probability of no deal is simply the complement, 36%.

Of course, if you think cross-party agreement and perfect whipping are merely "probable" (70%), then the probability of a deal through this route dips to ~50%, and the probability of No Deal rises accordingly.

To translate back into English: a path to a deal therefore requires a successful agreement between the leadership of the two main parties, no or very few defections on the Conservative side; a near-biblical commitment to abstention on the part of Labour MPs; and for all this to happen in the next forty-eight days. This is extremely demanding.

I realise that 40% is a terrible number to give: it's the number you give when you want it to appear that you have taken something seriously without indicating it as the most likely outcome.

This figure isn't free of consequences — if you believe me, you can do as I have done and bet on different forms of No Deal Brexit. Consider it a form of insurance against this disastrous-seeming but increasingly probable outcome.