Mar 9, 2020
Mar 9, 2020

How often does a bookmaker balance their book?

Bookmaking in a nutshell

A common misconception is that a bookmaker’s job is simply to “balance the book” to guarantee profit. While a bookmaker will often manage their risk across all the available options in a market, this might not always be the case. Read on to find out how often a bookmaker balances their book.

Last year I wrote an article examining odds setting from the perspective of the bookmaker, focusing on the favourite-longshot bias. My finding was that it will often be optimal for a rational, profit maximising and risk minimising bookmaker to introduce a favourite-longshot bias into their odds.

I wanted to write another article looking at odds setting and variance from the bookmaker’s perspective. In this article I examine the likelihood of a bookmaker balancing their book (profiting regardless of outcome) on any given market, as well as optimal odds setting strategies when bettor biases are introduced.

Balancing the book

A bookmaker isn’t always seeking to balance their book. Given their pockets will be deeper than most clients’, bookmakers will generally welcome variance, especially when they think they have superior predictions. The belief that a bookmaker is looking to balance their book minute to minute, hour to hour, or even day to day is generally an oversimplification.

Let’s assume two bookmakers want to offer odds on a coin flip. They both recognise the true probability is 50% for heads and 50% for tails. One wants a 5% margin, the other a 2% margin. They set their odds at 1.90 and 1.96 respectively.

One hundred square bettors decide to make an equal sized bet. They don’t know the true probability and therefore the outcome they end up backing is assumed to be random (50:50). To get an idea what this looks like from the bookmaker’s perspective I simulated this situation 5,000 times for each bookmaker.

Their range of positions before the coin is flipped can be visualised in the chart below. The 2% margin book is on the left, 5% on the right. A bookmaker’s potential position can be read vertically. For example, after laying 100 bets, the most extreme position for the 2% margin bookmaker was -33.3% if the coin landed Heads, or +37.3% if it landed Tails. This position occurs when the split of wagers is 68:32.

At the bookmaker with 5% margin, their book was balanced in 37% of the simulations. At the 2% margin book this was just 15%. How does this balanced book percentage look after 500, 1,000 or 5,000 bets?

 No. of bets 2% Margin 5% Margin 100 15% 37% 500 31% 75% 1,000 46% 89% 5,000 84% 100%

As we would expect, the more bets that a bookmaker can lay, the more likely they are to balance their book. We can see how much more difficult it is for a low margin book (even knowing the true probability and without any sharps in the market) to balance their book on a given market.

Exploiting bettor biases

Let’s now assume the 2% margin bookmaker has some information regarding the betting preferences of its client base, and predicts that there is a 60% likelihood that a bettor will back Heads. Where should they price the market?

Let’s continue assuming there are no sharps in the market. If the bookmaker holds the odds at heads 1.96 and tails at 1.96, their expected return remains constant at +1.96%. However, the possibility that they balance their book disappears if they lay any more than around 100 bets. Here’s what their range of positions would look like after 5,000 bets. The chart represents 5,000 simulations. Tails is, of course, the preferred outcome for the bookmaker.

In this scenario, the bookmaker has priced the odds efficiently but won’t balance their book because of bettor ‘irrationality’. If the bookmaker wants to maximise its likelihood of balancing its book, they would price the market off the predicted proportion of wagers. With a 2% margin distributed proportionally this would be somewhere around heads at 1.63 and tails at 2.45. If they lay 5,000 bets at these odds, their range of positions will look like this.

The bookmaker has increased their chance of balancing their book to 84%, while their expected return remains the same at +1.96%. We can see how a bookmaker who only takes bets from squares will be incentivised to offer inefficient odds, more closely reflecting their client’s irrationalities than the true probability. By operating in this way, they obtain the benefit of lower variance without having to sacrifice expected return.

Where they decide to price the market depends on their risk appetite. Continuing the example above, bookmaker expected return is plotted for different odds implied probabilities in the chart below.

The expected return will be highest when the odds are set at the implied probability midpoint between the true probability (50% here) and the predicted proportion of wagers on that outcome (60%). The bookmaker can maximise their expected return by setting the odds for heads at 1.78 (55% implied probability). What does variance look like with a market of heads priced at 1.78 and odds of 2.18 for tails?

This example shows that the incentive will always exist to shorten the odds for an outcome that a soft bookmaker believes will attract an inefficiently high proportion of wagers.

Next let’s consider what happens to the likelihood of balancing a book if the bookmaker sets odds efficiently and either ignores or doesn’t identify a bias in their clients’ preferences. Assume they set efficient odds (1.96 for both heads and tails) but the likelihood that a bettor bets on heads is unknown. In the chart below I have plotted this percentage over the range from 45% to 55%.

If proportion of bets laid strays slightly from the true implied probability (50%), the likelihood of balancing the book diminishes rapidly. For example, if a bookmaker lays 1,000 bets and the likelihood of laying a bet on heads is 46%, their chance of a balanced book reduces from 45% (at 50% likelihood of laying heads) to just 3%.

We can begin to see how challenging it is for a low margin bookmaker to balance their book in any given market. As the odds move away from even money this task only becomes more difficult.

Bookmaking in a nutshell

This analysis was predicated on a few strict assumptions, namely knowledge of the true probability, an accurate assessment of bettor preferences and a lack of sharp punters. Relaxing these assumptions will likely make book balancing even more difficult.

In reality, a bookmaker won’t know the true probability, bettors will bet differing amounts, and at Pinnacle, sharps will attempt to punish what they believe are inefficient odds. The addition of sharps will limit a bookmaker’s ability to exploit squares by offering odds that differ from their estimate of true probability.

Whether they ultimately set odds that diverge from what they believe to be ‘efficient’ will depend on a number of factors. These include the relative risk appetites of the bookmaker and the sharps, the proportion of square to sharp money, as well as the timing of each groups’ participation in the market.

This is bookmaking in a nutshell - a cat and mouse game of quantifying uncertainty, predicting bettor behaviour, and managing variance over time. On the one hand, nuanced and multifaceted, on the other, simple and predictable.

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