# On Brexit, Bookies, and Probabilities

I was, perhaps fortunately, not working with any client exposed to the Brexit risk. So when asked for an off-the-cuff opinion on which way the vote would go, I used a quick method of determining risk: I looked at the odds the bookies were offering. Here's why that went badly wrong...

Looking at the odds offered by bookies is often a good way of determining probabilities: the favorite generally wins. But that didn't work here. The final odds being offered on a British exit of the European Union were in the range 3/1 to 6/1, suggesting that the probability of the Brexit vote winning was in the range 14% to 25%. (If you're not familiar with odds being given in this form, there's a good explanation of the conversion of odds to probability here). The final odds can be found here.

Unlikely events do happen, but that wasn't the problem here. This was something else and, if I'd thought about the issue more deeply, I should have realized that the betting odds might be a bad proxy for the outcome of the vote.

To understand what went wrong, we need to understand a little more about bookies and bookmaking… .

[My apologies to professional bookmakers for this gross over-simplification. If I have made any major mistakes, please let me know.]

The objective of a bookie is to make money. To do this, a bookie has to ensure that the amount he (or she) will pay out is less than the amount bet regardless of the outcome of the event being bet on. He can't always do this, but it's important to realize that this is the objective. The bookie knows he isn't smarter than everyone else. If he relied on the idea that his assessment of probabilities was better than that of his customers he would be a gambler, not a bookmaker.

So the objective of a bookmaker is to set up a situation as follows:

Proposition Odds
Offered
Amount Bet Payout If
Team A Wins
Payout If
Team B Wins
Team A Wins9/10£1,000£900-£1,000
Team B Wins9/10£1,000-£1,000£900
Bookmaker's Profit£100£100

In this ideal situation the bookie makes the same amount of profit whatever happens.

But there is a problem. Gamblers can place a bet on either side of the proposition. Suppose gamblers think that Team A has 25% chance of winning, and Team B has 75%. With these probabilities, rational gamblers will immediately bet on Team B. This situation may quickly develop:

Proposition Odds
Offered
Amount Bet Payout If
Team A Wins
Payout If
Team B Wins
Team A Wins9/10£10£9£10
Team B Wins9/10£100£100-£90
Bookmaker's Profit£91-£80

Clearly our bookmaker now has a problem. He is now gambling on the outcome of the event and will lose if Team B wins. He can either stop taking bets on Team B or adjust the odds so that more people will bet on Team A.

Assuming he adjusts the odds, the situation should stabilize to something like this:

Proposition Odds
Offered
Amount Bet Payout If
Team A Wins
Payout If
Team B Wins
Team A Wins2.6/1£500£1,400£500
Team B Wins0.6/3£1,500£1,500-£400
Bookmaker's Profit£200£200

With this arrangement, a rational gambler sees neither proposition as a viable bet and the bookmaker makes a profit. There is no incentive for anyone to change the status quo. Crunching the numbers, the implied "collective wisdom" of the gamblers is that Team A will win between 17% and 28% of the time, and Team B will win between 72% and 83% of the time, which is what we would expect.

## And Then There Was Brexit…

For sporting events this can work reasonably well. The final odds are a reasonable proxy for the combined beliefs of professional gamblers.

So what went wrong with Brexit?

• There were a large number of amateur gamblers. Amateur gamblers bet on the side they want to win (e.g. their favorite team), rather than making a rational choice about which side of a proposition to bet on.
• People who had more money with which to bet (people who were doing well under the status quo) were more likely to favor Remain; people who were less well off (those not doing so well under the status quo) had less money to bet and were more likely to vote Leave.
• There wasn't a great deal of communication between the two groups, so people who thought that Remain should win were likely to talk to a lot of other people who also thought Remain should win. Even a rational gambler can make a poor probability assessment in this situation due to confirmation bias.

So if those favoring Remain average a £100 bet, and those favoring Leave are average a £10 flutter, what happens?

Proposition Odds
Offered
Amount Bet Payout If
Brexit Wins
Payout If
Brexit Loses
Brexit Wins6/1£100-£600£100
Brexit Loses1/10£1,000£1,000£100
Bookmaker's Profit£400£0

The bookies adjusted their odds according to the bets. The final odds at Ladbrokes were 1/10 Remain, 6/1 Leave, suggesting incorrectly that the probability of a Brexit was between 9% and 14%.

The actual proportion who voted for Brexit was 52%.

## TL;DR

• Bookies' odds are a poor proxy for probabilities when the bulk of gamblers are not likely to bet rationally, and when gamblers favoring one particular outcome can be expected to be more wealthy or more likely to bet than those favoring the other.

Was Brexit the right decision? It took 40 years for 52% of voters to decide on balance that it would be better to leave the EU. It may take another 40 years to decide if that was really the right decision.

2 November 2016

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