ANALYTICS MADE EASY
How to Compute Football Implied Probabilities From Bookmakers Odds
A comparative review
Introduction
When we are interested in betting there is always a time where we deal with probabilities. These probabilities can be derived from a mathematical model using data analysis or we can directly imply them from the bookmaker odds. Today we are conducting a study with the latter.
The dynamics of the odd market
Let’s take a step back to the basics. Odds are a reflection of the price punters can be ready to bet on an event. As stocks market, this price is mainly driven by supply and demand. When they are enough bet on an event (i.e. liquidity), we can assume that the odds contain all the information available (injured players, weather, teams form, and so on) that make the price fair.
On top of the market price, there are bookmakers' margins. Without margin, betting would be a null-sum game, the losers pay the winners, money goes from one pocket to another. But this ideal world does not exist, bookmakers take a piece of the cake. So odds are effectively the market fair price minus the bookmaker margin.
The bookmaker equilibrium
To understand that let’s use a simple example: tossing a fair coin. This experiment has two outcomes, tail or head. The chances to get one or the other is 50%, this is the true probability of the event. What would be the fair odd to offer as a bookmaker? Without margins, the answer is 2. Indeed, if you play that game repeatedly, neither the player nor the bookmaker would make money in the long run.
The fair odd is the odd at which none of the participant can make money on the long run. The money invested is redistributed, losers pay winners.
Unfortunately, the true probability is often unknown and has to be estimated. That can be done by a mathematical model for instance but also by quoting a market. Let us explain.
In theory, bookmakers always have to adjust their odds in such a way that losers are paying the winner. The quantity of money bet on each possible outcome has to be the same. This a weighing scale game played by the bookmaker where the odds have the role of the weights. If an odd is attractive, more money will flow on it and vice versa.
Assume the bookmaker does not know that the coin probability is 50% and he starts quoting head 5 and tail 1.25 with no margin. If you know the true probability (and the fair odd) as a punter, you will bet on the head and no one will bet on the tail.
Now the bookmaker faces a problem. If the result is head, he will have to pay from his poker as no one bet on the tail. To avoid this situation, the bookmakers need to change the odds.
The bookmaker has to make the tail more attractive. For instance 1.33 for the head and 3 for the tail. Now punters will go on the tail and the bookmaker will adjust again. Eventually, the bookmaker odds will converge to the fair odd of 2 for both tail and head as the punters are adding more information because they know the true probability.
In sports betting market, no-ones knows the true probability so odds are constantly moving.
The crowd effect
Now we understand why bookmakers are adjusting quotes, let’s take another example with the coin. Assume the bookmaker knows the true probability and quotes 2 for both events. For some reason, punters love head and 2/3 are betting on it while 1/3 bet on the tail. The bookmaker still needs to adjust and will change the odds accordingly. Eventually, the odds will converge to 1.5 for the head and 3 for the tail. Using the fair odd formula we obtain a 66% chance for the head and 33% for the tail, far away from the true probability.
When the market is liquid, odds are not reflecting the true probability but they reflect the proportion of punters that are taking a bet on each possible outcome. Often, these proportions are a good estimate of the true event probabilities.
Margin adjustment
We stated that betting is a sum null game but it is not true. Bookmakers take a margin, punters are using their services after all. The easiest way for them is to slightly change the odds.
Back to the coin example, a bookmaker would probably quote head 1.95 and tail 1.95, slightly below the fair odds. In this situation, the losers still pay the winners, but slightly less than they should and the difference goes into the bookmaker's pocket. The bookmaker can also adjust his margin differently. So he could quote 2 for the head and 1.95 for the tail and still make money as long as they are the same proportion of money bet on each side.
Margins are not fixed and only the bookmakers know what is the margin on each side of an event.
Risk Adjustment
The margin also represents the uncertainty (the risk) a bookmaker faces around an event. The bookmakers tend to increase their margin if they do have not enough punters (no liquidity). If all punters bet on the head, no matter the odds the bookmaker will offer on the tail, the bookmaker risks losing a lot of money. Early punters will have an odd of 2 while the crowd effect will making it converge to 1 eventually.
To cover this risk bookmakers can try to buy back odds before the event starts, at a discount price. For instance, any punter that bought the odd of 2 will be offered a buyback at 1.5. The bookmaker will save 0.5 in this case. This is what happened in 2016 with Leicester winning the English championships.
Why bookmakers do not have the same odds?
This is a simple question to answer now. Each bookmaker will have a different amount of bet on each event's outcome and have a different margin. As a consequence the odds are different. But in the case where everyone could have access to any bookmaker, the difference would vanish as the punters always go for the highest odds. The bookmaker with the highest odd would have to adjust and odds among them will eventually turn similar.
What are the implied probabilities?
So far we have seen how the true probability of an event can be related to the odds and why odds are moving. It is important to understand that implied probability is different from true probability. Implied probability is what the market thinks the true probability is, so it is an estimate of the true probability.
The implied probability is derived from the odds given the margin. Concretely we have
Since margin and liquidity are different from a bookmaker to another, implied probability also does. To it make sense to speak about a specific bookmaker implied probability rather than as a general concept
But it is not that simple to find. Indeed the margin is unknown and can be different for each of the results as we have seen in the tail head example. Only the bookmaker knows its implied probability as he can remove its margin.
For the rest of us, we just have to make sure that the sum of implied probabilities among the possible outcomes is 1. The problem is there are a thousand ways of doing it, making diverse assumptions on where bets are placed or how margin is distributed across the outcomes for instance. In what follows, we are presenting four methods¹ to compute the implied probabilities.
The Multiplicative Method
The multiplicative method is probably the most known and easy one. The inverse odd are simply rescaled by the sum of all inverse
The Additive Method
The additive method simply removes the same proportion from each inverse odd. It is also very easy to compute but probably wrong as there are few chances that margins are equal between outcomes
The Power Method
The power method uses a power coefficient to compute the implied probabilities
The constant k is found using an iterative method such as the sum of implied probabilities equals one.
The Shin Method
The Shin method is less popular and more complicated. It is also an iterative method based on a correction term z that corresponds to the proportion of informed punters.
The code to compute each of the methods is available on Github and a calculator is available here.
Comparing the methods on real data
This section compares the four methods using 1x2 odds collected from several bookmakers among multiple leagues. For each set of odds, we will run the four methods to find the implied probabilities.
The log loss (ℓ) will measure the quality of the probabilities. The closer to zeros the best. Given the implied probabilities of home wins, away wins, draw and the match result, the log loss for a given match is given by the following formula:
We compute the average log loss of each method to determine which method is the best.
The data
The data used for the comparison are fairly simple. We have collected odds from 7 bookmakers where we had the most of the data in over 900 competitions since the 1st of January 2018. The bookmakers are 10Bet, 1xbet, Betfair, Pinnacle, TitanBet, WilliamHill, and Bet365. Only odds snapped at least 45 minutes before the match starts are retained and we have filtered bad data points. We only kept matches where we have odds data for each of the bookmakers. Finally, we have 63861 sets of odds in our dataset equally distributed among bookmakers. The dataset is available upon request.
Overall result
First, let’s show the overall table of average log loss for each of the methods.
We observe that the methods are not dramatically different from each other. There are no statistical major differences but on average it seems that the multiplicative is the least predictive.
Clarke et al. (2017) found the multiplicative method was also less predictive for other sports.
Both Shin and the power method are very close in terms of predictive power.
Results per bookmarker
An alternative way to look at the results is to average the log losses per bookmaker. We have approximately 10000 matches for each of the bookmakers so we can compare them. The next table shows the results
We observe again that Shin and the power methods give the best results depending on the bookmaker: 10Bet, Betfair, Pinnacle, and bet365 have the best predictive results using the Shin method while the power method has the best results for 1xbet, TitanBet, and WilliamHill.
Results suggest that the implied odd method is different from one bookmaker to another which tends to support that bookmaker’s odds are driven by different margin and crowd effect.
Results per max odd
When there is no clear favorite in a match, odds are often more homogeneous. In these cases, we would expect each method to give similar results. Looking at the max odd per match is easy to see if there is a clear favorite. To see if it has an impact on the choice of the method we conducted the following experiment. For each match, we pick the max odd. Then we form 4 groups based on the 25%, 50%, and 75% quantile. For instance, the first quantile contains all matches where the max odd is in the range 0, 3.45 included. For each group, we average the log loss and the results are displayed in the next table
When the max odd on a match is under 3.45 (no clear favorite) all methods are equivalent. There is no benefit of using iterative methods like the power method or the Shin method. Margin seems equally distributed over each outcome.
In the last quantile, when odds are above 5.3 there is an advantage to use the power or the Shin method over the two others as the predictive power has increased by almost 1.5%. It seems logical that the margin will be lower on the favorite. Indeed assuming the odd on the favorite is close to 1, the margin would probably be close to 0 and both multiplicative and additive methods would be wrong.
Conclusion
Understanding how the quoting market is working for sports is essential to apply mathematical analysis. This article presents four methods to derive the implied probabilities from the 3 ways market odds.
Empirical analysis shows that Shin and power methods are offering the best predictive power. Depending on the bookmaker, a method can be better than the other, but overall the multiplicative method seems to be the less predictive.
However, the is no statistical evidence that one method gives the best results, even if our results are aligned with Clarke et al. (2017). Other markets could be explored, especially those with multiple outcomes like the match score, but we let that for another article.
A calculator for each method is available here.
[1] S. Clarke, S. Kovalchik and M. Ingram (2017), Adjusting Bookmaker’s Odds to Allow for Overround, American Journal of Sports Science 5(6):45.
