Surebet mathematics

Nowadays, there are two types of odds systems: the fixed odds system and the exchange odds system. For both systems the mathematics behind the surebet will be shown based on an example. Check the differences in oddssystems page, if you are unclear about the difference in odds systems or want to know how bookmakers and betting exchanges earn money.

Bookmakers operate on margins, making it theoretically impossible to find arbitrage opportunities at the same bookmaker, errors not included. The margin of the bookmaker is dependent on the outcome of the event, the quoted odds and the percentages of total bets on each outcome. This lead to different strategies among bookmakers, resulting in different odds. Bookmakers could try to forecast game outcomes or they could decide to price their odds based on the expected volume of total number of bets on each outcome. In order to create the right odds, bookmakers employ odds compilers who have knowledge of the specific event and have a good feeling about the most likely distribution of bets among the odds. In order to make use of arbitrage, bettors must make use of the differences in odds set by different bookmakers on the same event. The goal is to reverse the margin to the benefit of the bettor, using the highest available odds. Check the biases in odds page for more information about where differences in odds come from.

Fixed odds

Arbitrage opportunities in a fixed odds system are available when the following equation is satisfied:

Equation 1: 1 / highest odds outcome 1 + 1 / highest odds outcome 2 + … + 1/ highest odds outcome n <1                                                

Where n is the number of outcomes a sports event can have.

To make everything less abstract an example will be introduced.

Nadal and Federer play the final of Wimbledon. The bookmakers Unibet and Bwin are both offering the possibility to bet on this match, however the odds compilers of the different bookmakers have set different odds. The following pay-out table is available:

When these numbers are put into the formula stated at equation 1 we get the following:

Equation 2: 1 / 1.47 + 1 / 3.50 = 0.966 < 1  

This implies that there is an opportunity for a surebet available when a bet is placed on a win of Nadal at Unibet and a win of Federer at Bwin. How much exactly can be earned with a surebet can be calculated with the following formula:

Equation 3: 1 / (1 / highest odds outcome 1 + 1 / highest odds outcome 2 + … + 1/ highest odds outcome n)        

When the formula stated in equation 3 is used for the previous example the following can be obtained:

Equation 4: 1 / ( 1 / 1.47 + 1 / 3.50) = 1.035 = 3,5% arbitrage profit  

In order to reach this profit, specific amounts have to be bet on each bookmaker on the different outcomes. To calculate which percentage of the total stake exactly must be wagered on each outcome at the different bookmakers the following formula can be used:

Equation 5: (((1 / (1 / highest odds outcome 1 + 1 / highest odds outcome 2 + … + 1/ highest odds outcome n)) / highest odds outcome i)     

Where highest odds outcome i stands for the highest odds offered for the specific selected outcome. When this formula is used in the previous example the following equation shows:

Equation 6.1: (( 1 / ( 1 / 1.47 + 1 / 3.50) / 1.47 = 0.7042 = 70.42% of the total stake         

Equation 6.2: (( 1 / ( 1 / 1.47 + 1 / 3.50) / 3.50 = 0.2958 = 29.58% of the total stake   

Building forward on the previous example, in the case a bettor decided to use a total stake of €100,- and he faces no transaction cost he could make a total profit of €3,50 no matter the outcome of the event by placing €70,42 on Nadal at Unibet and €29,58 on Federer at Bwin. This is known as a surebet or an arbitrage bet.

Even within an efficient betting market this kind of arbitrage opportunities could arise, since bookmakers are trying to balance their own book and not the combined sports betting market book. Furthermore, biases in odds like the sentiment biases, differences in views of odds compilers and bookmakers trying to attract new customers with high odds on a specific outcome are common reasons arbitrage opportunities exist in the sports betting market. The likelihood of these opportunities is rising, because the competition is getting more intense and odds comparison websites like racingpost.com, oddschecker.com and oddsportal.com are increasing in popularity, causing average margins to get lower.

Moreover, the incoherence in fixed odds is also caused by the lack of objective means for assessing the probability of the outcome of  a sport event. Even a perfect understanding of probability theory would be of little help determining the odds. This makes it unlikely odds are formulated based on perfect probabilistic analysis. Most odds compilers working for the large bookmakers are early school leavers that have shown good results at local betting offices and are given some in-house training. The reaction of a managing director of Ladbrokes, one of the leading online bookmakers, on the question if he had ever considered hiring graduates with a statistical education was: “God Forbid”.

It is important to note that in the above mentioned equations risk is not included. Bets could change in between the time the wager has been set at one of the bookmakers, but not yet at the other. Furthermore, a bet at one bookmaker could get cancelled by the bookmaker, while the other bet does not get cancelled and keeps being locked in. These are some of the risks bettors pursuing an arbitrage strategy should keep in consideration.

Exchange odds system

The mathematics behind surebets with betting exchanges are almost the same as with the normal bookmakers. The only difference is that you first will have to calculate the true odds, which are the odds after commission. Once you have the true odds you can use the same equations as mentioned above.

You will have to pay a commission over the winnings of your bet, changing the true value of the odds. In this example we will use a commission of 6,5 percent. Therefore the true odds will change. You can calculate the true odds by using equations 7.1 and 7.2. Equation 7.1 is for backing and equation 7.2 is for laying a bet. Laying a bet means that you bet on a certain event not to happen.

Equation 7.1:  1 + ((odds - 1) x (1 - commission / 100)

Equation 7.2:  1 + ((odds - 1) / (1 - commission / 100)

Equation 8.1:  1 + ((1.32 - 1) x (1 - commission / 100) = 1.299

Equation 8.2:  1 + ((1.33 - 1) / (1 - commission / 100) = 1.353

Tip: If you want to see how much you will exactly get when you place a bet on Betfair, go to Display options and check on Display net of commission and Display “what if” figure.

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