How do Bookmakers make money?

"The safest way to double you money is to fold it in half" Kin Hubbard

"I like to play Blackjack. I'm not addicted to gambling. I'm addicted to sitting in a semi-circle." Mitch Hedberg

Gambling is a risky business, in the long run everyone loses. Well not everyone. Bookmakers always seem to do alright. How is it that bookmakers make so much money out of gambling then? Is it through their superior wit and street smarts, or is there something else going on? It turns out that the method used by bookmakers involves a lot less insight and risk than you might think. The process is mathematical, and is guaranteed to turn a profit, and actually quite interesting.

Making a Dutch Book

Let's start with an example, suppose I am a bookmaker offering odds on the Super Bowl coin toss. We all know that for a fair coin, the probability of getting a heads is 50%, and the probability of getting tails is 50%.

We might suppose therefore that we would set our odds at 1/1, meaning for a bet of £1 if you win, you get £1 back plus your original £1. This means that you would double your money if you win (which happens about half the time) and lose all your money if you lose (which happens about half the time). Over time, if you made lots of bets of £1 at these odds, you would expect to break even.

Let's suppose though, that for reasons unknown to us, far more money is being bet on heads than tails. For example, suppose £1,000 is put on heads, and only £500 is put on tails. We can then examine our position under two scenarios, one where heads wins, and one where tails wins.

We have taken in £1,500 in total, and our outgoings are either £2,000 or £1,000. Making us a profit of £500 half the time, and -£500 half the time. Given each outcome is equally likely, over the long run if we keep offering odds like these we will expect to break even.

We can do better than this though. We can actually set the odds so that we break even every single time, not just in the long run. In order to see this, let's suppose that we change the odds so that they are 1/2 for heads, and 2/1 for tails. So that means, for every £2 bet on heads, we will pay out £1, plus the original £2 bet, and for every £1 bet on tails, we will pay out £2, plus the original £1 bet. Under these new odds, we would then end up with the following position:

We see now that because of how we have amended the odds, we don't actually care who wins. We will always pay out the same amount. The Bookmaker can then adjust the odds slightly, so that they will always pays out less than they have taken in. For example, they might offer 5/11 for heads, and 9/5 for tails. Under these odds, not only will the Bookmaker not care who wins, but they will always make a small margin on all bets placed. The exact figures in this case would be:

So we see that the Bookmaker has a foolproof system of always making money from gambling, and plus, they don't even need to be good at predicting who will win.

Irrational odds?

Isn't it crazy to offer odds of 2/1 on a coin toss though? We all know​ that the actual odds for a fair coin should be 1/1! The answer is that as soon as people see these odds, they should start betting on tails, and by supply and demand the odds for tails will start to move down, and the odds for heads will start to move up.

The bookmaker themselves might also decide to take a position at these odds and effectively bet money themselves by allowing the payout to be skewed towards a 50/50 split, giving their payout a positive Expected Value.

The first time I read about this system of bookmaking was in the book "Financial Calculus: an introduction to derivative pricing" by Baxter and Rennie, which I was reading in preparation for the IFoA ST6 exam. Baxter and Rennie brought it up because Bookmakers are actually undertaking a form of arbitrage, similar to the series of  notional trades used when deriving the price of futures contracts. You can see the similarities by simply noting that both the bookmaker and the derivatives trader are acting so as to not have any exposure to the actual value of the underlying event. By doing this, they don't actually need to take a view at all on what the expected outcome is, but they can instead exploit the relative values of different parts of the market.

What margin to online bookmakers charge?

I had a quick look at some odds offered by online bookmakers on a couple of events. One of the events I looked at was UFC 218. BestFightOdds.com gives a comparison of the odds offered by various betting websites. This is mainly to help people pick which website to bet on in order to get the best odds, but we can use it to compare the margins that the different websites charge. Assuming an equal bet is placed on each fight on the card, I calculated the following margins for the websites listed by BestFightOdds:

They range from a low of 2.62%, up to about 6%. The individual margin on each fight varies quite a bit, and also I suspect each website will run different margins on different sports and different events. The reason for this is that some of the higher margin websites might offer more incentives, and more free bets than the lower margin websites., plus they might be more established and spend more on advertising allowing them to charge more and still retain sufficient numbers of customers.

The fact that websites run margins does mean that if you are going to make money gambling you need to first beat the margin before you can even break even. Let's say you're really good at gambling and you can out-predict the market by 2%, you would still lose money overall gambling on any of these websites, because you also need to factor in the margin (also called the vig) that the websites charge.

Sources:

(1) BestfightOdds.com:
www.bestfightodds.com/events/ufc-218-holloway-vs-aldo-2-1368