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.