Bayes says "no" to no-deal

One of the questions most frequently asked of me is what are the chances of a no-deal Brexit? In common with most analysts I tend to give an answer couched in subjective probability terms. This has the advantage that it is my view – nobody can say it is definitively wrong. Others may not agree with it, but they have the right to apply their own subjective assessments. However, this does strike me as a bit imprecise and led me to wonder whether it might be possible to derive a more accurate data-driven assessment of the odds of no deal.

One way to approach the problem is in terms of Bayesian statistics. In simple terms, Bayesian statistics assesses probability in terms of the degree of belief in an event. This contrasts with the more traditional frequentist school of statistics which represents probability as the number of times an event will occur based on an infinite number of representations of the process (I looked at this issue in more detail here). Bayesian statistics has always struck me as a sensible way to approach problems like Brexit where belief plays a big role and where the frequentist approach, which is based on the assumption that an event can be repeated, is unrealistic.

Bayesian statistics is based on Bayes’ Theorem which describes the conditional probability of an event based upon prior beliefs (or information) and which is written thus:

P(A | B) = P(B | A) . P(A)

  P(B)

This says that P(A | B), which is the probability of event A conditional on event B, depends on the probability of the event B given that A is true (the so-called posterior probability), and the prior probabilities of A and B.

To put this into Brexit terms, let us assume that event A is a no-deal Brexit and B is the event Remain. We are interested in the probability of a no-deal Brexit, P(A). We can rewrite this:

P(A) = P(A | B) . P(B)

P(B | A)

or

P(No-deal Brexit) = P(No-deal Brexit | Remain) . P(Remain)

P(Remain | No-deal Brexit)

Using data from the website What UK Thinks, we can use survey data to provide some of the evidence. Based on the question of how people would be likely to vote in a second referendum, we can derive a value for P(Remain) which is currently 44% (or 51% once we strip out those voters who either do not intend to vote or who have reported as “don’t know”). We proxy the value for P(Remain | No deal Brexit) by looking at the survey evidence which asks respondents to choose between no-deal and Remain. Faced with this choice, 44% of respondents indicated that they would vote Remain (54% once we strip out don’t knows). Similarly, we estimate a value for P(No deal | Remain) by looking at the proportion who would vote for no-deal if the alternative is Remain (38%, or 46% after adjusting for “don’t knows”).

Putting these numbers together gives a constrained probability of no-deal of 44% (where the constrained probability is derived by stripping out “don’t knows” and constraining the remaining categories to sum to 100%). The unconstrained probability, where we do not adjust for the “don’t knows”, comes out around 36%. The good news is that however we slice it, the numbers come out at less than 50%; the bad news is that they are higher than my subjective probability of 30%.

To the extent that politicians are looking at the polling numbers to assess whether a no-deal policy makes sense, this Bayesian interpretation of the evidence suggests that there is more support for a no-deal Brexit than is often supposed. However, it is not high enough to push through with it in the event that it proves to be economically disastrous, since more than 50% of voters do not support it and the backlash from the disaffected half is likely to be severe. One caveat is that some of the surveys have not been updated for a few weeks and therefore do not reflect any possible change of stance since Boris Johnson became prime minister. Moreover, Bayesian probabilities change as more information becomes available, so these are not set in stone by any means.

Bayesian statistics are the big thing these days but as the pollster Nate Silver pointed out, “under Bayes' theorem, no theory is perfect. Rather, it is a work in progress, always subject to further refinement and testing.” The prime minister’s new chief adviser, Dominic Cummings, pointed out in a blog post in 2017 “Rationality is more than ‘Bayesian updating’”. But cold hard statistics do catch up with you eventually and the evidence suggests that support for a no-deal Brexit is limited. If I were a politician it would not be the ground on which I would want to fight a battle – irrespective of what the prime minister says these days.

Don't bet on it

It is a truism in the gambling industry that the house always wins (although Donald Trump famously bankrupted his Atlantic City casino more than once). Being a bookmaker is generally viewed as a licence to print money although they don’t always get it right. One of the more famous examples in recent years was Leicester City’s Premiership win in 2016, despite having been priced as 5000-1 outsiders at the start of the football season, which cost bookmakers £25 million. It is in this light that we should treat the bookmakers odds for the Conservative leadership campaign, nominations for which close next week.

Just to put the numbers into context, the bookmakers are offering odds on 116 candidates, of whom 20 do not sit in the House of Commons whilst four are not even members of the party (one of them being Nigel Farage). This should make us a little bit suspicious as to the accuracy of the odds that are being quoted. At the time of writing, the bookies are offering odds of 2-1 on Boris Johnson making it to Downing Street (a probability of 37.5% derived from 24 different quotations) whilst second-favourite Michael Gove is being quoted at 4-1 (probability of 12%). Dig a little deeper and you find that the cumulated probability of the top six candidates sums to almost 100%. Given that there are 13 declared candidates , it is pretty clear that the sum of the implied probabilities exceeds 100%. Indeed, across all 116 candidates it sums to 181%. People are often surprised that this should be the case but this is to miss the point of what the bookies odds are telling us. 

Bookies odds should be treated as a payout ratio rather than as the actual probability of winning. After all, bookmakers’ objective is to make money from the volume of money placed on wagers rather than a rigorously objective assessment of the likely outcome. One of the ways which they do this is to take a slice of each bet in the form of a commission charge. In market terms, we can think of this as a bid-ask spread between the price the bookmakers are prepared to accept and the price at which they will pay out. In this case, however, the bookies appear to be charging a huge margin of 81% between the payout ratio and the true odds of the outcome (which by definition are limited to 100%). Ahead of the 2018 World Cup finals, the sum of probabilities across all participants was 115% which implies a much more reasonable bid-ask spread of 15%. But to see why this is the case, we need to consider some basic betting arithmetic and how this is affected by sample size.

The only thing we can say for certainty about the published odds is that they are designed to ensure that the bookies make a profit. The decisive factor determining the odds is the weight of money in favour of one or other bet. Imagine a case where there are 50 punters each paying £1, and 40 choose outcome A with a payoff of £1.2 and the remaining 10 choose outcome B with a payoff of £4.9. The bookmaker broadly breaks even in both cases (in outcome A, outlays are £48 and in the case of outcome B they are £49 - both less than the £50 of revenue). But if the balance shifts, with 30 people opting for outcome A and 20 for outcome B, the bookie makes a bigger gain in the event of outcome A (50 - 30 * 1.2 > 50 – 40 * 1.2) but will lose money in the event that B materialises (50 - 20 * 4.9 < 0). In order to reduce the losses on outcome B, the bookmaker is forced to reduce the outlays to £2.5 in order to break even (see chart) – in betting parlance the odds against have shortened. This has nothing to do with the fact that the bookie believes option B is now more likely. It simply reflects the weight of money switching to option B necessitating a change of odds to minimise losses.

The odds are also affected by the bookies’ need to make a profit. If, in our first example, the bookmaker targets a 10% return, they need to reduce the odds on outcomes A and B from 1.2 and 4.9 to 1.125 and 4.5 respectively which of course raises the implied probability (scenario 1a in the chart). Matters become more complicated when we extend the number of options: If our 50 punters can choose from 20 different outcomes, the sum of probabilities across the whole range of outcomes rises. It appears as though this is where we are in the Tory leadership race now: A long tail of outcomes quoted at long odds has raised the sum total of probabilities across the whole field. It is this combination of setting odds in order to minimise losses, together with the commission charged in order to make a given return whilst being spread across a wide field which gives the appearance of a very wide bid-ask spread.

If we were to constrain the bookies odds to sum to 100% and normalise the quoted outcomes appropriately, the odds on Boris Johnson taking over the job widen from 2-1 to 4-1 with Michael Gove widening from 4-1 to 7-1 and Andrea Leadsom (third favourite) from 6-1 to 11-1. What is interesting, however, is that the favourite for the top job almost never wins the crown. In the 54 years since the Party leadership competition was opened up to an election, rather than emerging as some sort of backroom deal, only once (2003) has the favourite won. And Johnson knows from bitter experience that the path to the top does not always run smoothly. We should treat the bookies odds with caution.