Bookmaker’s Margin and Arbitrage Betting

Calculate Bookmaker’s Margin

Betting companies are making a great profit due to their margins, which means that the gambler is expected to lose in a long term an amount equal to the (Total Betting Amount) X (Margin). Let’s see how to calculate the bookmaker’s margin by taking as an example the football match “Man Utd vs Arsenal” and the odds of Bet365 which is a famous betting company (30 Sep 2019).

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As we can see the odds are 2.3, 3.4 and 3 for Man Utd, Draw and Arsenal respectively. To calculate bookmakers margins with decimal odds, all you need to do is divide 1 by the odds for each outcome in the market, and sum together. This sum minus 1 is the bookmaker’s margin. Thus, we can easily calculate the margin as follows:

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Outcome	Odds	1/Odds
Man Utd	2.3	0.434782609
Draw	3.4	0.294117647
Arsenal	3	0.333333333
Total		1.062233589

Hence, the margin is \(1.062233589-1= 0.062233589 \approx 6.22\% \)

Calculate Bookmaker’s Probabilities

The Bookmaker first estimates the probabilities of each outcome and then adds the margins. The estimated probabilities are the (1/Odds)/Total. Finally, the “fair” odds (i.e. without margin) would be the 1/(Estimated Probabilities). Let’s calculate those figures:

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Outcome	Odds	1/Odds	Estimated Probs	Fair Odds
Man Utd	2.3	0.434782609	0.409309791	2.443137255
Draw	3.4	0.294117647	0.276886035	3.611594203
Arsenal	3	0.333333333	0.313804173	3.186700767
Total		1.062233589	1

As we can see, the Bookmaker estimates the probability of Man Utd to win to be 40.9% and the fair odd would be 2.443 times, but he pays back 2.3 times.

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From the Fair Odds to the Market Odds

The process that the Bookmaker follows is, 1) define the margin, 2) estimate the outcome probabilities, 3) calculate the fair odds, and 4) finally, apply the margin to the fair odds in order to get the market odds. The market odds are calculated as (fair odds) / (1+margin). In our case the (1+margin) is 1.062233589, so let’s calculate the Market Odds.

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Outcome	Odds	1/Odds	Actual Probs	Fair Odds	Market Odds
Man Utd	2.3	0.434782609	0.409309791	2.443137255	2.3
Draw	3.4	0.294117647	0.276886035	3.611594203	3.4
Arsenal	3	0.333333333	0.313804173	3.186700767	3
Total		1.062233589	1

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As we can see the “Market Odds” are the same as the “Odds” as expected. In case you got confused the Market Odds = 2.3 = 2.443137255 / 1.062233589.

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Calculate Bookmaker’s Mispricing

The Bookmakers can make a mistake in their estimates, however, they have this margin that we mentioned above which protects them from any “mispricing” in their odds. Let’s say that a gambler, estimates his own probabilities after applying Machine Learning Algorithms or even based on his experience and his gut feeling. Let’s assume that a gambler estimates the following probabilities and its corresponding odds (1/Probs) for the example above.

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Outcome	Probs	Estimated Odds
Man Utd	0.4	2.5
Draw	0.28	3.571428571
Arsenal	0.32	3.125

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As we can see, although in all outcome events they did not agree with the Bookmaker, however, there is not any “mispricing” since the gambler’s estimated Odds are higher than the market odds (e.g 2.5>2.3, 3.57>3.4, 3.125>3) which means that the fair price would be let’s say 2.5 but the bookmaker pays 2.3. Let’s see another example where there is a “mispricing”.

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Outcome	Probs	Estimated Odds
Man Utd	0.38	2.631578947
Draw	0.28	3.571428571
Arsenal	0.34	2.941176471

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In this example, there is a “mispricing” in Arsenal’s outcome, since the Bookmaker pays 3 times where the gambler estimates as a fair return to be 2.914 times, so he pays more! Thus, the gambler should go for Arsenal, although he does not believe that this is the more likely event to occur (34%). Again, the “mispricing” is based on the assumption that the “Gambler’s” estimated odds are more accurate than that of the bookmaker.

Note: The Bookmaker’s margin changes when we are dealing with multiple bets.

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Arbitrage Betting

Arbitrage Betting is an example of arbitrage arising in betting markets due to either bookmakers’ differing opinions on event outcomes or errors. When conditions allow, by placing one bet per each outcome with different betting companies, the bettor can make a profit regardless of the outcome. Mathematically arbitrage occurs when there are a set of odds, which represent all mutually exclusive outcomes that cover all state space possibilities (i.e. all outcomes) of an event, whose implied probabilities add up to less than 1

This is impossible to happen in one betting company, and if this happened, your bet would be canceled based on “terms and conditions”, but let’s say that you can play on different companies. So, the arbitrage will exist if the sum of 1/odds is less than 1. Let’s consider the following hypothetical odds.

Outcome	Odds	1/Odds
Man Utd	2.6	0.384615385
Draw	3.5	0.285714286
Arsenal	3.3	0.303030303
Total		0.973359973

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In this case we have an arbitrage of \(arb = 1- 0.973359973 =0.026640027 \approx 2.66\% \). Since we have an arbitrage, what is the best strategy in order to make a guaranteed profit, in other words, what are the weights that we should place in every outcome in order to win for every possible outcome?

The optimum weights are the (1/Odds)/sum, i.e for Man Utd even is 0.384615385/ 0.973359973 = 0.395141977. Let’s calculate the “weights” for each outcome.

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Outcome	Odds	1/Odds	Weights
Man Utd	2.6	0.384615385	0.395141977
Draw	3.5	0.285714286	0.29353404
Arsenal	3.3	0.303030303	0.311323982
Total		0.973359973	1

If there is an arbitrage the guaranteed profit that you can make is 1-1/sum(1/Odds) , which is in our case \(Profit = 1- (1/0.973359973)=0.027369141 \approx 2.7369\% \)

Let’s assume that we invest 1000$ in this game. The amount that we should place for every outcome is as follows (column amount).

Outcome	Odds	1/Odds	Weights	Amount
Man Utd	2.6	0.384615385	0.395141977	395.1419774
Draw	3.5	0.285714286	0.29353404	293.5340404
Arsenal	3.3	0.303030303	0.311323982	311.3239822
Total		0.973359973	1	1000

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If the outcome is:

• Man Utd, then we will get 395.1419774 x 2.6 – 1000 = 27.36914129$
• Draw, then we will get 293.5340404 x 3.5 – 1000 = 27.36914129$
• Arsenal, then we will get 311.3239822 x 3.3 – 1000 = 27.36914129$

Hence, no matter the outcome, we will make a profit of around 2.7369% and this is the definition of the arbitrage.

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Arbitrage in Practice

In practice, it is extremely difficult to find Arbitrage Opportunities, and there is also a risk to be banned from the betting companies. Below, we outline some difficulties.

Limited Opportunity In Limited Time

In the Internet age, in most cases, it only takes up to 15 minutes until an arbitrage opportunity disappears. Spatial distance has no meaning on the Internet. This is, of course, an issue if you are hoping to make regular profits from arbitrage betting.

High turnover, low profits

Another issue is the diminutive size of typical returns. In almost every arbitrage betting opportunity you will earn a relatively small profit from each transaction, a return of more than 3% on your investment will be a rare event. So while this gain is almost certain, you have to consider the rare nature of arbitrage. This means you need a lot of capital to invest in each opportunity, to justify even the high expenditure of time.

Be careful with high stakes

Even if you have a large amount of money to invest, another problem soon arises. Almost all other bookmakers accept large bets only when they are checked and waved through by a manager. So there is a real risk that one of your bets will not be accepted.