My estimates of the implied probability of a “leave” outcome in the EU referendum using the bookies odds from Oddschecker.com have attracted much comment. Some critics seem to have a weak understanding of the nature of prediction markets. For example, the argument is put that probabilities derived from the betting odds are of no value because punters are not a representative sample of the population. However, it should be obvious that those betting on the outcome of an event do not need to be a representative sample of the population – they are predicting, not voting.

To be precise, punters are staking their own money on the outcome of an uncertain event. For them, a rational strategy is to use all of the currently available information to form their own estimate of the likelihood that the event will occur. That information can come from any source – opinion polls, past history, the media etc. They can expect to make a profit, on average, if this likelihood differs from the odds being offered by bookmakers. Of course, this does not mean that all bets pay off: just ask anyone who didn’t bet on Leicester City to win the premiership.

Bookmakers revise the odds that they offer in response to the weight of money being placed by punters because they need to make a profit to stay in business. If, on the basis of the information available to them, punters are on average accurately predicting the odds of an event, then bookmakers will offer these odds to avoid making a loss. This strategy means they will at least break even over a large number of gambles.

To those who remain sceptical, I would suggest that they read Wolfers and Zitzewitz (2004) or Arrow et al. in Science (2008).

So far I have been estimating the probability that the outcome of the referendum on June 23^{rd} will be a vote to leave the EU. To some, this may be an unfamiliar way to present the outcome of a political contest: opinion polls gave a single (point) estimate of the *majority* for or against BREXIT at the time they are taken. The bookies odds provide an estimate of the *probability *of a “leave” (or “remain”) majority when the vote takes place by adding together the probabilities for each possible majority that will result in a “leave” outcome. If these are greater than 0.5, then a “leave” outcome is more likely. The reverse is true if the probability is less than 0.5. Currently this probability is hovering around 0.28.

However, bookies have also been taking bets on the *size* of the majority. It is possible to use this information to provide a more direct comparison between the bookies odds and the opinion polls. Further, one can show that the odds currently being offered on different possible majorities are consistent with the estimated probability of a “leave” outcome of 0.28.

Figure 1 shows the implied probability distribution of differing “remain” votes (current on June 3) derived by averaging the odds being offered by different bookmakers for different sizes of “remain” votes. It takes the odds being offered on a “remain” vote of less than 30%, 30%-35%, 40%-45% etc. and converts these into a probability distribution where the probabilities some to 1. Unlike opinion polls, which only offer a point estimate of the outcome, this approach allows a complete probability distribution to be formed, allowing, for example, the calculation of the chance of a “decisive” outcome that is so large that politicians are unlikely to revisit it for the foreseeable future.

**Figure 1: Probability Distribution of “Remain” Votes in the EU Referendum**

Source: oddschecker.com and own calculations

The most likely outcome is a “remain” vote between 50% and 55% (shown in red). The chance of this occurring is just greater than 0.25. This is clearly consistent with current estimates of the “remain” vote from opinion polls, although the 50%-55% vote for “remain” is only marginally more likely than the 55%-60% outcome.

How is this consistent with the probability of a “leave” outcome? From Figure 1, a “leave” outcome occurs when the size of the “remain” vote is less than 50%. The likelihood of this event, based on the bookies odds is 0.28, which is the sum of the probabilities for all of the outcomes (shown by arrows) that give a “leave” majority. This consistency between the overall vote and the votes for different majorities is expected: otherwise there would be an opportunity to make money without taking any risk by correctly designing bets on the overall outcome and the individual majorities.

One final point – for the last 40 years, the probability that the UK would be part of the EU two years hence was 1 – certainty. The probability distribution shown in Figure 1 illustrates how far from certainty this outcome now is. For businesses and others seeking to plan for the medium to long term, this is a form of risk that they are duty bound to consider and which is likely to hold back, rather than accelerate, investment plans.

References

J Wolfers and E Zitzewitz ‘Prediction markets’ Journal of Economic Perspectives (2004) 18 107–126