## The First Bookie

Posted in Football, mathematics, Sport with tags , , , , , , on April 24, 2019 by telescoper

I read an interesting piece in Sunday’s Observer which is mainly about the challenges facing the modern sports betting industry but which also included some interesting historical snippets about the history of gambling.

One thing that I didn’t know before reading this article was that it is generally accepted that the first ever bookmaker was a chap called Harry Ogden who started business in the late 18th century on Newmarket Heath. Organized horse-racing had been going on for over a century by then, and gambling had co-existed with it, not always legally. Before Harry Ogden, however, the types of wager were very different from what we have nowadays. For one thing bets would generally be offered on one particular horse (the Favourite), against the field. There being only two outcomes these were generally even-money bets, and the wagers were made between individuals rather than being administered by a `turf accountant’.

Then up stepped Harry Ogden, who introduced the innovation of laying odds on every horse in a race. He set the odds based on his knowledge of the form of the different horses (i.e. on their results in previous races), using this data to estimate probabilities of success for each one. This kind of `book’, listing odds for all the runners in a race, rapidly became very popular and is still with us today. The way of specifying odds as fractions (e.g. 6/1 against, 7/1 on) derives from this period.

Ogden wasn’t interested in merely facilitating other people’s wagers: he wanted to make a profit out of this process and the system he put in place to achieve this survives to this day. In particular he introduced a version of the overround, which works as follows. I’ll use a simple example from football rather than horse-racing because I was thinking about it the other day while I was looking at the bookies odds on relegation from the Premiership.

Suppose there is a football match, which can result either in a HOME win, an AWAY win or a DRAW. Suppose the bookmaker’s expert analysts – modern bookmakers employ huge teams of these – judge the odds of these three outcomes to be: 1-1 (evens) on a HOME win, 2-1 against the DRAW and 5-1 against the AWAY win. The corresponding probabilities are: 1/2 for the HOME win, 1/3 for the DRAW and 1/6 for the AWAY win. Note that these add up to 100%, as they are meant to be probabilities and these are the only three possible outcomes. These are `true odds’.

Offering these probabilities as odds to punters would not guarantee a return for the bookie, who would instead change the odds so they add up to more than 100%. In the case above the bookie’s odds might be: 4-6 for the HOME win; 6-4 for the DRAW and 4-1 against the AWAY win. The implied probabilities here are 3/5, 2/5 and 1/5 respectively, which adds up to 120%, not 100%. The excess is the overround or `bookmaker’s margin’ – in this case 20%.

This is quite the opposite to the Dutch Book case I discussed here.

Harry Ogden applied his method to horse races with many more possible outcomes, but the principle is the same: work out your best estimate of the true odds then apply your margin to calculate the odds offered to the punter.

One thing this means is that you have to be careful f you want to estimate the probability of an event from a bookie’s odds. If they offer you even money then that does not mean they you have a 50-50 chance!

## A Problem of Sons

Posted in Cute Problems with tags , , on January 31, 2019 by telescoper

I’m posting this in the Cute Problems folder, but I’m mainly putting it up here as a sort of experiment. This little puzzle was posted on Twitter by someone I follow and it got a huge number of responses (>25,000). I was fascinated by the replies, and I’m really interested to see whether the distribution of responses from readers of this blog is different.

Anyway, here it is, exactly as posted on Twitter:

Assume there is a 50:50 chance of any child being male or female.

Now assume four generations, all other things being equal.

What are the odds of a son being a son of a son of a son?

## Betting on the Supreme Court

Posted in mathematics with tags , , , , , , , on December 6, 2016 by telescoper

This week the UK Supreme Court is hearing an appeal by HM Government against the judgment recently delivered by the High Court which was that the UK Government must seek the approval of Parliament before it can invoke Article 50 of the Lisbon Treaty and thus begin the process of leaving the European Union. You can watch the proceedings live here. I had a brief look myself this morning but as I’m not a legal expert I found it rather hard to follow as it’s rather technical stuff. That wasn’t helped by the rather dull delivery of James Eade QC who was presenting the government’s case. Nevertheless, it is a very good thing that we can see how the law work in practice. I was surprised at the lack of gowns and wigs!

Although Eade seemed (to me) be on a very sticky wicket for some of the time, it’s impossible for me to come to any informed inference about who’s likely to win. Out of interest, to see what other people think, I therefore had a quick look at the betting markets. Traditional bookmakers (such as William Hill) are offering 1-3 (i.e. 3-1 ON) for the original decision being upheld so they’re clearly expecting the appeal to fail.

These days, however, I’ve started to get interested in other kinds of betting markets, especially the BetFair Exchange. This allows customers to act as bookmakers as well as punters by offering the option to “lay” and/or  “back” various possible bets. “Laying” betting means effectively acting as a bookie, proposing odds on a particular outcome. i.e. selling a bet.  “Backing” a bet means buying a bet. The exchange then advertises this to prospective bettors who sign up of they are prepared to stake money on that particular outcome at those particular odds. It’s very similar in concept to other trading services, e.g. share dealing. Matches aren’t always made of course, so not every bet that’s offered gets accepted. If that happens you can try again with more generous odds.

The advantage of this type of betting is that it represents an “efficient market”. Such a market occurs when all the money going into the market equals all the money being paid out in the market – there is no leakage or profits being taken. Efficient betting markets rarely exist outside of betting exchanges – bookmakers need to reap a profit in order to run a business. For example, though William Hill is offering 1-3 on the Supreme Court ruling being upheld, the odds they offer against this outcome are 12-5. These are not “true odds” in the sense that they can’t represent a consistent pair of probabilities of the two outcomes (as they don’t add up to one). In the case of an exchange market a bet laid at 1-3 is automatically backed at 3-1. These can then be regarded as “true odds”.

This is what the BetFair Exchange on the Supreme Court hearing looks like at the moment (you might want to click on the image to make it clearer):

The odds are given in a slightly funny way, giving the gross return for a unit stake (including the stake). In more normal language “4.3” would be 100-30, i.e. a £1 bet gets you £3.33 plus your £1 back. A bet on “overrule” at “4” (3-1) corresponds to a bet against “uphold” at 1.33 (1-3), reflecting what I was saying about “true odds”.

The first thing that struck me is the figure at the top right: £38,427. This is the value of all bets matched in this market. By BetFair standards this is very low. A typical Premiership football match will involve bets at least ten times as big as this. As in the court case itself there just isn’t very much action!

Apart from that you can see that the odds here are broadly similar with William Hill etc with implied odds around 3-1 to 4-1 against overruling.

Before you ask, I’m not going to bet on this myself. My betting strategy usually involves betting on the outcome I don’t want to happen. Although I think Parliament should be involved in Article 50 I am just happy that this matter should be left to our independent judiciary to decide.