How did Leicester city win despite 5000:1 odds? (Warning: for the stat savvy only)

Leicester City winning the EPL despite 5000:1 odds against winning has been such an extreme phenomenon that it is more extreme than the odds against Elvis Presley being found alive- 1000:1. At the beginning of the season, Leicester City were ranked 17^{th} out of 20 teams and barely managed to escape a relegation last year. Since 1992, the year in which the premier league was established, only the top 5 teams have won the League (22 times), except in 1994 by Blackburn Rovers and they too were runners up in 1993, which means they were amongst the expected teams to win. Leicester City’s whole team was bought at a price less than 1/10^{th} of the league’s most expensive buy.

The event is quite inexplicable and even more confounding considering the below two facts:

- Success doesn’t follow a bell curve and therefore it is even more difficult to win for a low ranked team:

By Central Limit Theorem, the mean of most natural phenomena follow a bell curve (normal distribution). Let me explain: If we take the mean heights of human population it will be a bell curve. This is because the height of individual A let us say is dependent on three random variables- nutrition, spend and weather.

Now these can be expressed as an equation:

Height = B1*nutrition+B2*spend+B3*weather+E, where B1, B2, B3 are the constants and E is the randomness (unexplained variance due to randomness). Central limit theorem says the 3 random variables are independent and have an additive effect on the natural phenomenon- height of an individual. Additionally, the mean height of these individuals put together results in a bell curve.

However, success follows log normal-distribution and not bell curve. That is, as shown like the picture above, whenever there is a competition among various individuals (or football teams), the resulting mean score of merit is distributed log-normally. That is most of the individuals will be in the lower end of the distribution and the successful few will be at the higher end.

This is because the factors contributing to merit are not additive, but multiplicative. For example let us say merit is explained by the following equation for an individual:

Merit = (B1*environment)*(B2*spend)*(B3*talent)+E. They are multiplicative because, an individual having higher spend and better talent will have a multiplicative effect on the merit, whereas the same individual will have an additive effect on the height (natural phenomena). It is like saying that if one is talented as well as does hard work then he is not 2(sum) times more effective but his effectiveness is a product of talent and hard work. The Central Limit Theorem fails in success because it is for independent (additive) random variables and as soon as the multiplicative effects happen the resulting distribution is no more normal and is log-normal (there is also a log normal Central Limit Theorem if you want to delve more into multiplicative effects)

Therefore, in EPL we obviously observe that the same top 4 or 5 teams win repeatedly, because they recruit better talent, spend more and also the victory effect of the previous seasons are all multiplicative to produce the resulting merit. The rest of the teams have no way of winning/getting to the right side of log-normal distribution since they already possess lower talent and lower scores and on top of that merit is multiplicative. So the victory chances of Leicester city are very low.

2. Law of large numbers:

The lower ranked teams might win one or two games (due to chance), but over an entire season of 38 games the individual random variables show their true colors and the merit gets averaged out. As a result the summation of the scores for merit for Leicester City will result in a quantity proportionate to their rank and they are bound to end at the bottom of the table most probably.

Now, despite the above 2 factors how did Leicester city manage to win?

The answer lies in two important factors:

1. Leicester city capitalized on the randomness factor E: Now, let us first define randomness (definition from information theory). For example: If a coin is flipped, there is no way of identifying whether it will return heads or tails with human eye. This is because the randomness (unexplained variance) is so high that we cannot predict it using human eye. However, if we use better tools and sensors, then it is actually possible to predict a random event like tossing a coin. The randomness associated can be decreased. In game theory randomness is like luck. In some of the video games, the developers purposefully inculcate randomness in order to increase the unpredictability of the games. Therefore, higher the E, higher the randomness/luck factor.

Chess for example has zero randomness if played by 2 computers because every move can be associated with definite odds. But, so is not the case when humans play. Humans can bear in memory only a definite number of future moves and after that, the game becomes unpredictable. Similarly, football has a certain degree of randomness. The data analysis has reduced randomness a lot (ex: the Money ball movie) though. The machine learning algorithms can include hundreds of thousands of variables and therefore in our equation the E (unexplained variance/randomness) gets converted into more and more explainable factors and becomes smaller. The simulation algorithms can run games with the opposing teams’ formations and strategies/tactics and therefore the luck factor is greatly reduced esp. with teams having better spending capabilities and data hacking capabilities. Therefore Leicester city did not have much randomness left in the game hereafter to produce unexpected outcomes.

In addition, Leicester city did not compete with regards to the possession variable (ranked 18^{th}) at all, which has been a major point of discussion among the competitors. It also seems logical to conclude that higher the possession, better the chances of scoring a goal. Well, Leicester city has turned that upside down we know. In terms of pass percentage they are placed 19^{th}. So they basically do high risk passes with high degree of randomness. They did not/could not compete with the variables that are usually explained to calculate merit of the teams because they know they cannot do that with the kind of talent and money they possess.

So how did they beat the giants in the game?

They capitalized on whatever randomness was left in the game, to the most. As soon as an opposition attack is thwarted, there is unlimited chaos and randomness in the game of football. Here the formations go out of shape. So all the tactical and strategic moves from the opposition are out of picture. The E is at the highest. And during this vulnerable chaotic situation Leicester masters the game and scores. Rest of the time (and also overall) they are the same below average team in terms of merit. But they hacked a way to increase the score line with the help of capitalising on the few randomness moments that are left in the game today (technology, I am sure will soon consume that too and make it predictable).

Leicester did this with the help of long passes (they have the highest) and sheer pace when other teams were focusing on regaining position and composure from chaos. Leicester’s game itself was modelled around creating (to some extent using 2/3^{rd} of the playing area only at any point of time, long risky passes) and capitalising on the chaotic situations the most (pace and high conversion %). Since most premier league teams work on short passes and possession oriented games, Leicester created new variables that explained the remaining E. There is a big lesson in how to defeat stronger teams/opponents here. Identify the E (randomness) and develop a strategy to score when it is high. There is no way to compete in the usually explained variables like possession. Considering Leicester’s talent, they obviously loose there and that is not their focus. But whenever E went high they capitalized the most and made it count to the score line.

Next time when you face a tougher opponent bear in mind the E he has left out and the E you can create and capitalize on in every situation.

2. Once Leicester found a way to capitalize on randomness and built the variables around the E, they also made sure the team was very fine tuned. A fine-tuned team again takes up multiple variables and creates a multiplicative (if not more) effect on the merit. Using these fine-tuned variables they were able to quickly move to the right of the log-normal table, just like they consistently moved into the D. Team spirit itself takes up many individual random variables and creates an output which has more than just an additive effect. The fact, that they ended up at the very extreme of the log normal table despite having low talent means they gelled the random variables really well to create a multiplicative effect on the merit outcome. The team manager’s role in building such a fine tuned team cannot be ignored.

And that is how Leicester won the league. By capitalising on the randomness and building a very fined tuned team around the same according to me. I would be very keen to hear any counter arguments on the same J