The Mathematics of Football

The mathematics of football

1) A few more general considerations

The task one faces when trying to get a mathematical “grip” on football is not an easy one. First of all, of course, one needs the necessary world view in order to be able to find a mathematical approach at all, or rather, to embark on the search. The basic worldview is that a football match is a random experiment with an indeterminate outcome. It is not actually predictable how a game will end. There may be favourites and underdogs, but even then it remains the same: there are only more or less probabilities for the occurrence of each possible outcome of this random experiment.

In particular, a football match, following the chapter on “How odds are created”, is a sport played by individuals, which provides a perfect example of the thesis that any random experiment conducted in practice is a mixture of perfect predictability and absolute chaos.

I am happy to provide another discussion, also at this point here: Absolute chaos” sounds rather unpleasant. However, in a mathematical sense, it only expresses unpredictability. If we take a football match in which, as the media always tell us, “in the end only the goals count”, i.e. the goals scored by both teams after 90 minutes, then we find that there are three possible outcomes to this random experiment: Team 1 has scored more goals and is therefore the winner (of this match), team 2 has scored more goals and is the winner or both teams have scored the same number of goals and the match is declared a draw (the special case of a cup match in which a winner may still have to be determined by extra time and a penalty shoot-out is not of interest to us here for the time being; we simply always consider the regular final whistle after 90 minutes).

Absolute chaos would therefore mean that we have no idea whether team 1 will win, the match will end in a draw or team 2 will win. So if we approach without any prior knowledge, we would first have to say, also borrowing from the dice example, “Since I don’t know any better, I assume all outcomes are equally likely.”

But the opposite of that would be “perfect predictability”. A result of this might look like this: “I know how the game will turn out afterwards. Germany wins.” (And that’s true, of course. But only because it’s Germany.)

The truth about this lies almost naturally in the middle. One part is chaos, here we should better call it, subordinated to coincidences. Another part is most certainly predictable. But the calculable part is only to be understood as calculable to the extent that the probability distribution can be shifted a little, and in the best possible way to reflect reality.

As a purely random experiment with three outcomes, it would require, force, a probability distribution of 1/3 for team 1 to win, 1/3 for team 2 to win and 1/3 for the draw (or draw). That would be non-predictability, then; chaos; the pure chance experiment; the LaPlace experiment. The shift in the distribution of chances would be in a direction like 50% win team 1, 30% draw and 20% team 2 or some other prediction deviating from the equal distribution.

It is obvious by individual and also team aptitudes (I remind you: a team is more than the sum of the individual players) that such a shift exists. There are clubs with greater financial resources, which is (can be) reflected in the club structure, training facilities, coaches and ultimately the players themselves.

The clubs are working to shift the odds in their favour, so to speak. It is even the intention not to achieve an equal distribution. There must always be the favourites (and their downfalls). The surprises add the spice. The game itself also holds enough contingencies that can influence the outcome of the game. Smaller clubs often strive (in their respective divisions) only to be “competitive”. Others are condemned to success. This pressure can also be paralysing at times. Becoming an overachiever is sometimes easy because there is no pressure. Sustaining that can be the much bigger task because successes create pressure of expectation.

To what extent this is sufficient for predictability, or how much this shift in the distribution of chances can be calculated, is the task I have dedicated myself to with my programme.

As far as measuring the ultimate quality of the forecasts is concerned, there is on the one hand a long-term financial result but on the other hand also a statistical method that is suitable for testing. This can be found in the chapter “Testing the quality of forecasts”.

—— Still to be revised (or deleted) from here onwards

If you take my background into account, the invention of the computer meant that one day I would use it to continue my childhood playing career in adulthood and use the computer for this purpose.

I may simply assume that I have a certain talent for numbers and computer logic. This did the rest, so that one day I almost inevitably got involved with the problem of how to get a mathematical grip on football.

The control parameters required for this were already apparent from early childhood, at least that I was aware of their existence. The most important ones are obviously the goal average and the home advantage. In addition, there is the playing strength, which I already tried to take into account as a child on the basis of the table positions. That’s not enough, but at least it’s a start.

2) The ultimate idea

The basic idea is to determine the probability of all possible outcomes of a football match. Here, the possible outcomes are quite specifically. 2:1, 3:3, 0:0, 4:2, these are football results. If you had them all, you would also have the probabilities that one team or the other would win or that the game would end in a draw.

In football, the essential characteristics are the goals that both teams score against each other in the match. However, unlike the fortune teller, I have tried to express this number of goals that the two teams will score as goal expectations. The fortune teller tries to read the outcome from his crystal ball, whereas I try to be as good as possible with the help of the goal expectations and to read the changed distribution of chances from them.

If you succeed in doing this and have the most realistic goal expectations for a game, then you are still faced with the next problem. This problem is to read off the probabilities for a win/draw/loss from the goal expectations. I will explain below how this can be done. But first I want to draw your attention to the consequences if you succeed.

Because: If you then know the starting probabilities for a single match, you can also use them to determine the starting probabilities for an entire tournament. Be it a World or European Championship or even a whole season. Of course, not only the Bundesliga but worldwide for all leagues that are recorded. Suitable leagues are those where you can get enough information to make an estimate for the participating teams.

3) The implementation

So we have two tasks: First, the goal expectations for a given match must be calculated as realistically as possible, but mathematically logical, using the parameters that are provided.
Once we have these, we have to convert these goal expectations into 1-X-2 probabilities, the “tendencies”.

Now I have to define a term again or make it unambiguous in its use: For example, the result of a match is 2:1, 4:0, 2:2, 0:3 or 1:1. These are the exact results. When compared, these give a “tendency”. I then call this tendency either tendency or 1-X-2, win – draw – defeat.

But how are you supposed to find out the probabilities for the results? The magic word here is “simulation”.

With the help of this simulation, one could find out the probabilities. Because: a simulation reflects reality as far as possible. The outcome is still subject to randomness. Even a high favourite can stumble, the question is always how likely this will happen. So the simulation is run not just once but 1000, 5000 times (computers can do this, and you wouldn’t manage to snap your fingers once at the same time). After a high number, you then see how often this or that result occurred. How often did the favourite actually win, how often did they only draw and how often did the underdog even win.

I’ll continue step by step, however, and we’ll first look at…

—— Still to be revised (or deleted) up to here

4) The parameters and calculation rule

When you approach a problem like this mathematically, there is always the question of parameterisability. Can the football be parameterised? The answer is, of course, yes. The only question is which parameters one declares to be significant. There are also different approaches here, I myself had already chosen at least one other way earlier (this was not sufficiently good, as I soon discovered) and even found two other approaches after the implementation of my currently still used system, both of which also have their strengths and weaknesses.

For the time being, however, I will only explain the one I am currently using, but I am also happy to mention its weaknesses in conclusion.

It is obvious to me that there are parameters that determine the distribution of chances in a football match. The question is what they are. Then there is always another question, which is: what are the general parameters and what are the specific parameters.

The general parameters that are quite obvious to me are the goal average and the home advantage. There is a long-standing goal average and there is a long-standing home advantage that simply emerges, which is obvious: the home teams win significantly more games.
The specific parameters are the playing strengths of the teams.

Now, what is the best way to express a team’s playing strength in a game where the goal is to score as many goals as possible while conceding as few as possible? The answer lies in this question:
By giving each team an offensive strength and a defensive strength. This offensive strength is best measured in an average number of goals scored per game. In other words, goals they will score on average. The defensive strength is expressed in a number of how many goals they will concede on average. Both numbers together make up their playing strength.

Now, first of all, the term “on average” needs to be explained. My system works best for league play, which, however, also accounts for the majority of sporting events. In this context, people also like to speak of “everyday life” for the professional footballer. So on average in this case means that a team’s goal expectations reflect the expectations against all teams in a league that they would have to score and concede in the course of the season in the first and second legs. The best way to explain this is, as usual, with an example: Bayern Munich had a goal ratio of 68:21 in the last season (2007/2008). The whole scored in 34 games makes at Division 2 : 0.62. So roughly their playing strength should correspond to these two values. Their offensive strength is about 2.0 (goals scored per game), their defensive strength is 0.62 (goals conceded per game).

To make the whole thing a little less dry, I asked my database out of curiosity how reality and prognostics related to each other in the case of Bayern Munich for the 2007/2008 season. And my database said the following: The sum of all expectations of Bayern’s games last season was 63.31 : 25.53. So in the offensive I assessed Bayern relatively well (to very well), in the defensive it was lacking: I didn’t give them enough credit, i.e. they conceded 4.5 goals less than my computer expected (it must just be the “Titan Kahn”, right?). But still. As a mathematician, I say “tolerable”. Someone would have to have made alternative predictions and shown me that they were better.

That’s how it is with every team. Their playing strength is expressed as goals they (should) score on average against all teams. The same applies to the goals conceded. So these specific parameters are measures of playing strength. How many goals do I expect them to score, how many do I expect them to concede.

If a specific match takes place now, then in theory at least we have two match strengths for the time being. In principle, the home advantage applies league-wide. The only thing is that I have established from the beginning (and established before) that the home advantage must also be maintained individually. So there are teams that tend to be strong at home and those that tend to be weak at home or strong away from home. However, this can also change over the course of a season.

But now we have two playing strengths, the specific parameters, and two home advantages, also specific. And we have a goal average and an average home advantage. These are the general parameters. Now these values have to be offset against each other to predict the goal expectations for a specific match.

Note: The parameters are all non-rigid. The match strength parameters only ever express the match strength at the current time. The same applies to the specific home advantage. They have to be constantly maintained (i.e. “updated”, see section “Update of playing strength”).
By the way, the general parameters are also updated by adjusting the specific parameters.

When searching for a formula, one must always observe certain basic conditions: The formula must be valid for the simplest cases. So a simple hypothetical case is this: Bayern plays, first on a neutral pitch against an exactly average team. Then, of course, their own goal expectations must come out as the goal expectations for this match. So, the 2007/2008 season was quite a successful one for Bayern. They were above their long-term average, also in terms of the quality of the goal ratio. The long-term average is something like 2:1. They score 2 goals per game and concede 1 goal. So let’s take their playing strength as 2:1. If I now calculate the goal expectation for the game in question, the result must be 2:1.

The mathematician generally has a certain disease. That is expressing variables and parameters in abbreviations. Only it is partly useful and necessary to be able to write down a formula at all. However, as usual, I try to provide the intuitive reasoning as well. So the parameters are the following: Tew1, Gtew1, Tew2, Gtew2, DT. These denote in detail: Tew1 = Goal expectation team 1. Gtew1 = Goal against expectation team 1. The same for team 2. DT are the average goals, a general parameter that a team should score on average. So since there are two teams involved in a match it is half the goal average. DT is therefore = goal average/2.

Now how do we offset these parameters against each other? As usual, this is best explained vividly and made clear with an example:
To illustrate, we first calculate how many goals team 1 scores more/less than the average. This is expressed by the quotient of Tew1/DT. So if it is better than the average in offence we have a factor greater than 1, if it is weaker than the average the factor becomes less than 1. It is its ratio to the average team in offence. Second, we calculate how many goals team 2 concedes more/less than average. Why we calculate this is actually clear: The two values “how many goals more/less than the average team 1 scores” and “how many goals more/less than the average team 2 concedes” belong together. We want to find out the concrete goal expectation for the match and therefore first calculate how many goals team 1 will score in this concrete match (of course only as an expected value).

Second, we calculate the quotient for team 2. For team 2, however, we have to take into account their defensive strength. This defensive strength is calculated analogously to the goals scored by team 1 more/less than the average. So it is the quotient of Gtew2/DT. This expresses exactly what we want to know. In the same way, we get a value of greater than 1 here if the team is worse than the average (so it concedes more goals than the average) and less than 1 if it is strong defensively.

These two calculated values work in the same direction. If team 1 scores more goals than the average, the first value is greater than 1, if team 2 concedes more goals than the average, the value is also greater than 1. And the goals scored by team 1 in this game are the goals conceded by team 2. In this respect, the arithmetic operation that one must then use is also clear: one must multiply the values together. Two factors greater than 1: The product becomes even greater. So an offensively strong team against a defensively weak team logically results in a “goal festival”, but only in the expectation. Both factors smaller than 1 means: The value is further reduced by multiplication. But also correct: An offensively weak team meets a defensively strong team. How are they supposed to score goals? The value becomes small. Of course, it is just as correct when an offensively strong team meets a defensively strong team: One value goes above 1, one below 1, results in a value close to 1 in the multiplication, which is also correct. The reporters then like to talk about the teams “neutralising” each other. And they’re not even entirely wrong.

All right, after multiplying the two values we have a new factor. Depending on the offensive strength of team 1 and the defensive strength of team 2, this value is then large or small, greater than 1 or less than 1. This value expresses how many goals more/less than the average team 1 would have to score in this specific match against team 2. The calculated value must then be multiplied by DT. In this concrete match, I expect team 1 to score the goals more/less than the average that a team would have to score in the match.

Finally, I summarise all the calculation steps again and put extra (superfluous) brackets to keep the values apart: We calculate (Tew1/DT). Then we calculate (Gtew2/DT). Then we multiply these values together and get (Tew1/DT)(Gtew2/DT). And at the very end we have to multiply this by DT. Results in (Tew1/DT)(Gtew2/DT) * DT.

Unfortunately, the mathematician’s urge to simplify always prevails. This leads to the equally unfortunate realisation that one can “shorten” the final expression. You can “shorten” the final DT, which appears as a factor, against one of the DTs in the quotient, thus eliminating both completely. Mathematicians are not concerned about clarity. I also just want to mention it: The final formula is therefore: Tew1*Gtew2/DT. As simple as possible. Three parameters, two belong in the numerator, one in the denominator. That’s it. Finding formulas is a bit like baking pretzels. And I really can’t do that at all.

5) A few examples

If we use the values for the simplest case, the result is of course the right one:

The calculation goes like this: The long-term (last 10 years) goal average in the Bundesliga is 2.84 goals. So the average team has a goal expectation of 1.42:1.42. So the value DT is 1.42. Bayern has the values 2 and 1 for Tew1 and Gtew1, the average team has boringly the values 1.42:1.42. So the multiplication gives 21.42/1.42 = 2 and 11.42/1.42 = 1 for the expected goals (and goals against) for this match. So exactly the 2:1 that Bayern just has as a playing strength.

For another match, I calculate once again, but still on a neutral pitch, i.e. without the parameter of home advantage.

I’ll even print out a current table in advance so that you can check the values a little and perhaps make a better assessment:

Let’s take the match Werder Bremen – 1.FC Köln from 15.11.2008. The current goal average in the league is currently 2.94, which means that more goals have been scored recently than the long-term average. Werder Bremen’s match strength is 2.02:1.43, so they have scored a lot of goals, but also conceded a lot (because the sum of the values is 3.45, i.e. higher than the goal average). 1.FC Köln has had a very good season so far. However, as a promoted team, they are of course not yet as highly rated as their current position in the table. Their current playing strength is 1.31:1.43.

This results in an expectation of 2.02/1.47 for goals scored by Werder Bremen. 1.47 is DT, because goal average = 2.94, divided by 2 = 1.47. This is the factor that measures how many goals Werder score more than the average. The value 2.02/1.47 is 1.374. Cologne concede 1.43, so even slightly less than the average (note also their current goal ratio: 13-14, 14 is less than the average). As a factor, this is 1.43/1.47 = 0.973. Multiplying the factors gives 1.374*0.973 = 1.337. Werder score significantly more than the average, factor 1.374, Cologne concede less than the average, factor 0.973. It gives the factor 1.337 for this match. The value is smaller than the value for Werder alone, as Cologne’s value counteracts. Multiplying by DT gives 1.337 * 1.47 = 1.965. This is Werder’s goal expectation for this match (on a neutral pitch!).

Correspondingly for Cologne’s expected goals scored in this match: Their factor is 1.31/1.47 = 0.891. So they score significantly fewer goals than the average. Werder concede 1.43, although a lot (for a top team) but still less than average. Expressed as a factor: 1.43/1.47 = 0.973 (curiously, the same value as for Cologne’s goals conceded; pure coincidence). Multiplied out 0.891 * 0.973 = 0.867. So Cologne scores this factor less than the average in this match. Multiplication gives 0.867 * DT, so 0.867 * 1.47 = 1.274. We have the two values for this match out: Werder Bremen – 1.FC Köln (on a neutral pitch) results in a goal expectation of 1.965 : 1.274.

Werder remain favourites, despite their worse position in the table. I guarantee that every bookmaker would do the same.

The use of the home advantage parameter is analogous. It is only multiplied everywhere. Individually, because it can be different for the teams themselves. The average value is also taken into account. So the analogous question is: How many goals does team 1 score more (or less; the factor is always around 1; at 1 it is the average, less than 1 it is less, more than 1 it is the average) than the average home team and how many do team 2 concede more (or less) than the average away team. I’ll spare myself the formula here, though. It just gets a bit more complicated. For the treatment of the problem in principle, this does not bring any further enlightenment, I think.

6) Simulation

After having correctly calculated the goal expectations for a concrete game in this way, the question is how to derive estimates for the probabilities. I used my childhood memories and had the idea: simulate. The same thing I had always done back then. Only now the conditions were much better: maximum realism. But I also needed my football knowledge for that. And I did it in two ways: First, I divided the goal expectations over the 90 minutes. Every minute, practically every team had a chance for an attack. And their probability of scoring a goal was by dividing the calculated number of goals per team by 90.

Unfortunately, one finds that reality just doesn’t want to behave the way a little mathematician’s brain would prefer. The number of draws that resulted from this type of simulation was clearly too low. One sets out to investigate the causes. And here, for a second time, I was able to use my football mind, which I have acquired through observation:

The first, simplest and immediately obvious reason is that when a game is tied, the tendency to take risks decreases, purely intuitively speaking. After all, you have something to lose. Even if it is only the one point. At that time, by the way, the two-point rule still applied everywhere. So there was only one point more for a win compared to a draw.

So if a game is still tied, say, 20 minutes before the end, both teams gradually come to terms with the one point. This encourages the tendency towards a draw. My simulation was initially set up in such a way that the teams use their offensive and defensive strengths independently of each other. So you would have to imagine it as if both teams were making efforts behind closed doors to score a goal, or better still as many as possible. At the end, the doors are opened and they ask each other, “So, how many goals did you score?” and thus have the result.

In reality, the behaviour of the two teams depends on each other. So each team has the opportunity to react and adjust to the number of goals scored by the opposing team. This leads one to the second, perhaps not quite so and also not immediately, obvious reason for the tendency to draw: When a team is behind, it’s a tiny bit related to the feeling of panicking.

Panic really does give you superhuman strength, at least for the moment. And when you’re behind, when you’re in danger of losing, you can use a little of those powers. “Help, I’m behind, I’m in danger of losing, but now I have to make a double effort.” Or something like that.

Besides, by the way, there’s nothing more to get out of it for the leading team than they’ve already achieved. They are leading and want nothing but the final whistle. Tactically, any change can only benefit the trailing team. They have already lost. So in an emergency, the goalkeeper can also attack.

In the simulation, this had to be implemented realistically in such a way that I introduced a dampening of the goal probabilities when the game is tied and an increase in the goal expectations when the game is unbalanced. That is just the reality. You often hear comments like, “The game desperately needs a (first) goal to get going.” That’s how it is. Then something happens. However, when the equaliser is scored, the efforts are often reduced again. Certainly, there are individual differences. Nevertheless, through this intervention I have brought the draws to realistic values.

The probabilities of occurrence can now be determined by repeatedly running the simulation for each game. I used to work mostly with 5000 runs. With the possibility of simulating individual matches, one can now of course also simulate entire seasons. In other words, you can determine probabilities for long-term bets, such as “Who will be German champion” or “Who will be relegated” or “Who will be European champion” and so on. Here, each game is simulated only once, for an entire season or European Championship, then you have an answer for one run, who became German champion. Then this process is repeated, also 1000 or even 10000 times. Then you have a good estimate of how likely it is that Bayern will become German champion (or HSV).

7) Odds formula

As you can read in the chapter “How odds are calculated”, the odds offered on a sporting event are, of course, always calculated with a profit advantage from the bookmaker’s point of view. The bookmaker has to live on something (if you have any doubts about this, please read the book “Beruf : Spieler” by Dirk Paulsen). But how must this profit advantage be calculated correctly?

To explain the solution to this problem, I must first explain that there is a problem with this in the first place. If you have a probability of occurrence, you simply take the inverse and have the (correct, fair) payout ratio. All well and good. And if you calculate with profit, you just subtract something from the correct payout ratio, a certain percentage for example.

The simplest example: a bookmaker offers odds on a coin toss. Well, if you do it like Mike did against Jons at night in the chess café, there is no discussion about the payout ratio. 2.0. Equal money, done. The bettor has to make a living. So he says he will pay 1.95 on each side. The advantage for the bettor: although he gets a little less than would be “fair”, he can come up with a stake size that suits him. And hopefully he has no reason to doubt that he will get the money paid out if he wins.

Paying 1.95 on an event where the probability of occurrence is known with some degree of accuracy may seem intuitively neither exaggerated nor underestimated. It is realistic. On the Asian betting market, by the way, bets are regularly offered on “who will kick off”. However, this is not a pure coin toss. There are acting persons who influence the odds out of politeness or for other reasons. The payout odds are similar, but occasionally you get over 2.0. What’s the reason? No idea.

But in football matches otherwise, even if you estimate the probability at 50%, you would not pay out 1.95 as a bookmaker, at least. Because there is still the imponderability of the correctness of the assessment. So with an estimate of 50%, one would rather pay out 1.85. That is 15% less than correct, than the fair odds.

If you apply this procedure to other probabilities, i.e. basically subtract 15% of the fair rate from your estimate, you very soon run into the problem: If you estimate very small probabilities, for example 1% probability of occurrence, the fair rate would be 100. 100 – 15% = 85. You would have to pay a rate of 85 for the event that was estimated at 1%. I don’t know about you, but when I arrived at this consideration, I realised that one would never, ever pay it.

The reason then gradually becomes clear, at least why one recoils: it’s a very high rate, it’s unattractive anyway. You can only lose a lot and win nothing. But beyond that: what about even a tiny mistake in the calculation of the assessment? Maybe they got it wrong by a measly one percent, the truth is 2%. So correct, fair would be to pay 50, the reciprocal of 2/100, so 100/2. And one pays 85? That would be a huge mistake.

So very small probabilities are much more subject to the danger of misjudgement. Conversely, if you estimate 80%, a small error in estimation is hardly noticeable. Whether 81 or 79? Hardly any difference. So with high probabilities, with the stable values, one tends to pay even a little more than the fair rate minus 15% and with small probabilities considerably less.

So I set out to find a correct and reliable formula to solve the problem. And I found it. And the formula I found then is still in use today. It is mathematically absolutely flawless, and it is parameterised in such a way that it can still be adapted to individual wishes.

I am writing it down here. I know that its significance for the history of mankind is only slight, but nevertheless I felt a little like Einstein must have felt once for the time of research and discovery. I had a few prerequisites as to what the formula had to do, had a certain, rudimentary, previous education of a mathematical nature and set to work. So here is the formula:

ln((1/root(0.5 – Abs(0.5 – p) * qf)+1) / ln((1/root(0.5 – Abs(0.5 – p) * qf) * p + 1)

I admit that it is not too handy. Nevertheless, I can explain it very briefly: p is the probability for the event for which the odds are to be created. The parameter qf denotes the odds factor, which can still be set individually. So a bookmaker who wants to work with more profit has to keep this value low, someone who wants to calculate less percent profit can choose a higher value. Realistic values for qf are between 3 and 40.

The term 0.5 – Abs(0.5 – p) measures the distance of the value p from the centre, i.e. the distance to 50%. The only important thing is that an event with 80% is treated the same as one with 20%. By substituting it, one recognises it and the distance to 50% is also the same for both, namely 30%.

A root applied to values between 0 and 1 (and that is where probabilities lie) inflates them somewhat. For example, the root of 0.6 is 0.7745. The ln, i.e. the logarithm naturalis, simply must not be missing. The two terms in the numerator and denominator differ only in that the denominator is multiplied by p before the ln is applied to it. Multiplication by p causes the value in the denominator to become smaller. The addition of 1 ensures that the value before applying the ln is certainly greater than 1, because all previous values are positive and between 0 and 1. And the ln applied to a number greater than 1 always results in a positive number. So there are positive numbers in the numerator and the denominator. The numerator is greater than the denominator. As a result, the result from this formula is always greater than 1. And in this way it is suitable as a ratio.

An odds less than 1 would mean that the bettor would certainly lose. I would like to present a few more results to make the formula clearer.

I have created the diagram for probabilities between 12% and 88%. The reason is that the clarity is lost when you go up to 0 and 100%. You can no longer see anything.

The purple line represents the fair odds, i.e. the inverse of the probability. The blue line is the “pay rate”, which is calculated from the probability (i.e. the fair rate) using my formula. As you can see, the distance between these lines is getting smaller and smaller. With reliable, large probabilities, one becomes a little “bolder” and calculates with smaller profit yields. This is because of the more reliable estimation. This diagram was created with the odds factor 6.

To read out and interpret values: If the fair odds are 5, then one would be prepared to pay odds of 3.9 with an odds factor of 6. That is quite realistic. But it is a relatively large calculated profit. The calculated profit would be 22.6%. At fair odds of 2.5, the bookmaker would still pay 2.15. The advantage, also calculated as a percentage, would therefore be considerably smaller, namely only 13.2%. The condition is fulfilled to calculate a high profit with small probabilities, due to the greater susceptibility to error, and a small profit advantage with high probabilities, due to the greater reliability of the assessment.

Here in the diagram is the development of the profit advantage:

The jags get in there, by the way, because I’m already working with rounding. The computer would always calculate odds of 2.184 or 3.8726. You couldn’t write that down as a bookmaker. So the odds are rounded, the top two, for example, to 2.15 or 2.20 or 3.9 or 3.8. Otherwise, however, the progression is nicely recognisable, in my opinion. Small probability – big calculated profit, big probability — small advantage.

Here is another one with the odds factor 30:

The course of the curve does not change. The only difference is that this provider is, so to speak, bolder overall. The odds factor can be used individually. So if a provider knows one league very well, he can use a high value, for another a smaller one.

Now let’s look at the development of the calculated profit:

You can see that the calculated profits are all at a lower level. The prongs remain, or, one could also quite well say here: a prong sharper. And, to take the pun to the same level: The worse the round, the sharper the point. And I’ll also let you in on my omnipresent inner conflict: I had to fight two inner battles for this example of humourlessness, and this term already contains the solution to one of the two problems: The first was whether it would be funnier to write, the worse rounded the more pointed the zack. And the second, whether one can decisively increase the hilarity triggered by jokes by analysing them in depth?!

8) Playing strength update

So now the system is complete. Using the goal expectations, the probabilities for the match outcomes can be determined with the help of the simulation. Long-term forecasts can also be made. With the odds formula, we can even try our hand at betting, at bookmaking. Nevertheless, one question remains open with the system: How does one react to the actual results?

Obviously, it seems that an assessment of playing strength is related to the results. Good results also produce a good assessment, bad ones a bad one. But there are also developments. If I think a team is strong, but it keeps losing, there is somehow a need for action. Perhaps I have overestimated them?

There is also a rather obvious aspect: good results have a positive effect on self-confidence. You lose the fear of making mistakes. Even the morale of the whole team is improved. Suddenly you are friends with your comrades. Conversely, when results are bad, everything suffers. Instead of friends, you look for culprits. And so on. So, in short, you have to react to the results.

But how does one determine how to react in the first place and how strongly to react? The first part of the question can be answered as follows: One reacts by adjusting the goal expectations of the teams. Both offensively and defensively, of course. And how do you adjust them? Well, there is a goal expectation for each game and then a result. So there is a deviation of the result from the forecast. This is favourable for one team and unfavourable for the other. Both in terms of this goal expectation. So if, for example, goal expectations of 2.34:0.75 are calculated for a home match between Bayern and Bochum, then a 2-1 result is not favourable for Bayern in terms of goal expectations, even though the team has won (again). Their playing strength would suffer. They scored 0.34 goals less than expected and conceded 0.25 more than expected. So one would (have to) correct these two values in that direction.

The only open question now remains how strongly one should react. The criterion is and remains to achieve the best possible forecast for the following game. So, unlike in chess (see chapter “Elo system”), where forecasting is not in the foreground (but it should be, just by the way), here I am dependent on finding the best possible adjustment.

For this purpose I have used old data. Known results, arranged chronologically. I started with a baseline estimate and used it to forecast the games of the upcoming (first) matchday of a season. Then I made adjustments with a certain strength. In practice, this looks like dividing the deviation by a given value. So in the example above, you divide the 0.34 and the 0.25 by a certain value, say 20, and adjust the goal expectations for both teams in the right direction with the values you get. So 0.34/20 = 0.017 is subtracted from the offensive goal expectation for Bayern, and the same value is subtracted from the defensive for Bochum. This is because they have also received this number less and fewer goals conceded have a positive effect.

Then you forecast the next matchday with the newly received values. With the newly calculated expectations, you again get a deviation between the predicted values and the values that arrived. These errors, i.e. the total deviations, are added up for all matches. Then you get a result that you could call “total goal deviation”.

Then the whole procedure is repeated for the value 21, which means that the reaction to a result is somewhat slower and less pronounced. The errors are again added up for all games. And then this total goal deviation is compared with the previous result. And either 20 is better or 21. So gradually you find the best possible value in this way, namely the one where the total goal deviation is the smallest.

Nevertheless, it is not easy to find the optimal value. The main problem is the basic estimation of the teams’ playing strengths in the past. If you vary them, simply because you don’t know them exactly, then you get different optimal reaction times. But I have made several runs to find the best value and it turned out to be 30. So you could express it in two ways: 1/30 is the influence of the last result on the playing strength. Or just like this: The last 30 results determine the playing strength. That also sounds good.

9) Simulation substitution formula

Just to make it complete: The simulation delivers quite good results. The whole approach provides an overall structure that works reliably together. It’s just that one is always looking for improvements. The problem with a simulation: the results are not always the same. So if I have identical goal expectations for two games, the computer would calculate a probability of, let’s say, 53.48% of winning one game, and 52.09% another time. That can happen in a simulation.

So you look for a formula that represents as exactly as possible the numbers that would come out in a simulation. To replace the simulation with a formula. The problem is that, as described above, I have already made some refinements to the simulation. You try to represent reality as well as possible. And if you make a new observation, you might even find new tactics to reproduce it in the simulation. As soon as you have the formula, this possibility disappears. Or rather, you would then have to reactivate the simulation, try to interpret the results and find a new representation of these results with the help of a formula. And that’s not really what you do.

Anyway, one day I found a formula. It is essentially based on the Poisson distribution. In principle, the Poisson distribution is the Gaussian normal distribution for discrete values. Discrete here means: Only whole numbers. This is in contrast to the normal distribution, which allows all values, including any intermediate values. You could take the height of all adult men in Germany as an example. This is a so-called continuous distribution. Theoretically, one can be 1.83m but 1.83275m tall. In theory, such distributions always follow the normal distribution. There are many people of average height, a few very tall people and a few very short people. It accumulates around the mean and the number of people with a certain deviation from the mean becomes smaller the larger the deviation is.

A discrete distribution arises in football, for example, in the number of goals. You can only score 0 goals, 1 goal or 2 goals (it is curious here that you have to use the plural for the 0. But if we stick to German terms, it quickly becomes clear that the singular is reserved for the number one. Everything else is plural, even if it is the “uncount”), but not 1.5 or 2.73. Although these numbers are possible as expected values. So the Poisson distribution gives you probabilities for the specific numbers of goals for each team.

Hopefully, the example explains it better. So the following columns of numbers:

TorewM1	TorewM2
1.73	0.85
	Wkeit1	Wkeit2	akku1	akku2
0	17.73%	42.74%	17.73%	42.74%
1	30.67%	36.33%	48.40%	79.07%
2	26.53%	15.44%	74.93%	94.51%
3	15.30%	4.37%	90.23%	98.89%
4	6.62%	0.93%	96.84%	99.82%
5	2.29%	0.16%	99.13%	99.97%
6	0.66%	0.02%	99.79%	100.00%
7	0.16%	0.00%	99.96%	100.00%
8	0.04%	0.00%	99.99%	100.00%
9	0.01%	0.00%	100.00%	100.00%
10	0.00%	0.00%	100.00%	100.00%

These are the concrete figures for two absolutely realistic goal expectations of two teams in a match against each other. Team 1 has a goal expectation of 1.73 for this match, after calculating its playing strength and home advantage with the opponent’s strength (and away strength). The opponent, team 2, has a goal expectation of 0.85 after the same calculation. The probabilities of scoring a certain number of goals for each team are listed below. The columns akku1 and akku2 give the accumulated values up to the number of goals. So with 6 goals for team 2 or with 8 goals for team 1, the 100% has already been awarded. At least with the rounding of my computer. 8 goals are possible, even for team 2, but the probability of team 2 scoring 8 goals is only 2.9 * e^(-6), written out as 0.0000029. Approx. 3 millionths.

Here is the whole thing as a diagram:

You can see that the probabilities for up to 10 goals become so small at some point that, realistically, they can hardly occur. And who remembers the last 10:3 in the Bundesliga? Oh, it didn’t exist yet. 10 goals for one team did, even a maximum of 12.

So now we have two distributions, how often one team scores a certain number of goals and how often the other team scores a certain number of goals. In order to calculate how likely a certain result is, one could simply multiply the probabilities. So a 0:0 occurs when team1 scores 0 goals, probability 17.73%. That team 2 scores = goals is 42.74%. Multiplying the two values, which is allowed in the case of independence and is also the correct arithmetic operation, gives a value of 17.73% * 42.74% = 7.58%. The probability for the 0:0 would therefore be 7.58%. In the same way, we can calculate the probabilities for the other results.

If we then want to calculate how often team 1 has won, we add up all the values where team 1 has scored more goals. Likewise for the draw, by multiplying out all entries where the number of goals is the same and adding them up. Likewise for victories by team 2.

Again, more concrete: The probability for a 0:0 results from multiplying the values 17.73% * 42.74%. That is 7.58%. In the same way, you can multiply out all the results and get the following matrix (the matrix is only up to 5:5; but you can see from the sum columns that up to that point there are already 99.11% of all results).

	Record matrix
	0	1	2	3	4	5
0	7.58%	6.44%	2.74%	0.78%	0.16%	0.03%	17.72%
1	13.11%	11.14%	4.74%	1.34%	0.29%	0.05%	30.66%
2	11.34%	9.64%	4.10%	1.16%	0.25%	0.04%	26.52%
3	6.54%	5.56%	2.36%	0.67%	0.14%	0.02%	15.29%
4	2.83%	2.40%	1.02%	0.29%	0.06%	0.01%	6.62%
5	0.98%	0.83%	0.35%	0.10%	0.02%	0.00%	2.29%
	42.37%	36.02%	15.31%	4.34%	0.92%	0.16%	99.11%

By adding the upper half of the matrix we get the probability of victory for team 2, by adding the lower half we get the probability of victory for team 1 and by adding the main diagonals we get the probability of a draw.

We thus obtain the following values:
1 X 2 Sum
57.37% 23.55% 18.18% 99.11%

The remaining 0.89% are the results where one team (including those where both do) scores more than 5 goals.

Now, you may ask, what is the problem? Why didn’t he use such a simple method in the first place? The answer is the same as in the simulation: the problem is the draws. They occur too rarely even when using the Poisson distribution for the same reasons.

But, Pauli creates a problem for himself, Pauli solves the problem. I have made a redistribution on the main diagonal. A systematic and mathematically correct method. And depending on the desired draw frequency, the draws are adjusted accordingly. Responsible for this is the so-called draw factor. This must, as one easily discovers in practice, vary from league to league.

Even here, however, the necessary sensible considerations do not end. It is relatively obvious that a league with a lower goal average also produces more draws. In this respect, it is therefore not surprising that more draws occur in France than in Germany.

However, it is also a fact that the numbers that occur in practice cannot be due to the goal average alone. For this purpose, here are the comparative figures for the 2008 season:

Germany: Goal average arrived: 2.81
Expected: 2.75
Draw arrived: 25.49%
Expected: 24.72
France: Goal average arrived: 2.284
Expected: 2.198
Draw arrived: 30.52%
Expected: 29.84%

These numbers don’t look too spectacular. In France, there are practically half a goal less per match on average. Logically, this also means that more draws occur. But: In France I reckon with a draw factor of 0.89, in Germany of 0.93. So, apart from the lower tendency to score a goal at all, there is also a greater tendency to play for a draw. So is it possible that the willingness to take risks is also lower in France in general?

These are just examples here. The parameters also adjust for this by entering the result. So the draw factor as well as the goal average and home advantage are maintained and updated by the database, both individually and in general.