Complexity is a measure of how much variation there is in a system. Variation is a measure of how many variables there are and how many different values these variables can have. The more variables you have, especially when many of these variables can have many different values, the more complex the system is. A different way to explain complexity, is noting that the longer it takes to describe all the different options a system has, the more complex the system is. (Source: An Introduction to Complex Systems Science and its Applications by Alexander F. Siegenfeld and Yaneer Bar-Yam 2019)

Given how much you need to write about football before you have said it all (if that is even possible), football is a game with a very high level of complexity!

Our interest in complexity lies in our desire to deal with complexity. Cybernetics is about communication and control in man and machine (which is actually also the subtitle of the first book on Cybernetics by professor Wiener). As said, football is a very complex game and the question is how do we control this complexity. The first step to take control is to define the system and the environment. We are completely free to define anything in football as a system. As we have very little control over the opposing team, it is best to define our own team as the system and let the opposing team be part of the environment. The question then becomes how can the manager regulate the way the team (or the system) deals with complexity.

## Simplifying complexity

Obvious if you take both teams as the system, there are more variables than if you only look at your own team. More variables, means a higher level of complexity. But because we can’t control the opposing team directly, we chose to look only at our own team. In fact it is even easier to start with a team of only one player.

So let’s start with a single player. Imagine a match with only a single player. What can he do? Basically he can shoot the ball and run after it. Or he can dribble with the ball. There is not much else to do. So the complexity is relatively low. But as soon as we add another player, the complexity increases. Now they can pass to each other. One can give an assist as the other player scores. So team members increase complexity. Team members are variation amplifiers.

The more players we add to the team, the more complexity rises. In fact it rises exponential as all the possible different relations between the players add to the level of variation. A single team of eleven players without an opponent has the highest level of complexity. That sounds counter-intuitive. Doesn’t the opponent make the game more complex? The answer is: only if you make them part of the whole system. If the opponent is part of the environment, then the opponent is limiting the options of your team (the system in focus) and thus complexity decreases.

The opponent makes it more difficult to score. Obviously, It is much easier to score if there is no opponent on the field. Without an opponent it is very easy to score. But writing down all the options a team has, if there is no opponent, has become a much bigger task, than writing down all the options a team has when there is an actual opponent. Your opponent will diminish the options your players have. As such opposing team members are variation attenuators. Your opponent only increases the complexity if it is part of the system. But because we have excluded the opposing team from the system, we only look at what the opposing team means for the options our team has.

Now let’s add an opposing player to our experiment. Immediately, it becomes clear that in this situation the team has less options to score. For one, they can’t run the ball through the opposing player as that would be a foul. The team has to play around the opposing player. In fact, by adding a single opposing player we proof that opposing players are indeed variation attenuators. They diminish complexity. The more opposing players there are on the field, the less options our team has. The more our team is forced to play in certain predictable patterns. And the more predictable our team becomes, the more difficult it becomes to score. The reason why it is difficult to score in football is not because the game is too complex, the reason is that the game is not complex enough.

If you are not yet convinced than reverse our little thought experiment. Now we start we only one player for ourside against eleven opponents. Scoring becomes almost impossible as the eleven opposing players limit the option of our one player to next to nothing. Again, we can increase our odds by either adding players to our own side as they are variation amplifiers. With more players our team gets more options, complexity rises and our chance to score increases as well. Or we could decrease the number of opponents. With less variation attenuators our options increase and again complexity rises together with our chance to score.

### Real variables

Let’s see if we can calculate the level of complexity for a single player on an empty pitch. We will consider the following five variables (even though there are probably many more to consider):

- Position
- Direction
- Speed
- Timing
- With or without the ball

To calculate the complexity we have to determine how many different possible combinations of these five variables there. To do that we first determine how many possible settings each variable itself has. “With or without the ball” is easy as it is a binary variable with only two options: yes or no. For position we have to create a grid on the pitch. Let’s go for the smallest pitch allowed which is 100 by 64 meters. If we build a grid out of a single m2 then we have 100*64 = 6400 different positions on the field. For direction we have 360 degrees around us. But it is probably fine grained enough to use sections of 30 degrees so we can work with 12 directions as if we were to use a clock to determine our directions. Timing goes really fast in football so it is probably best to use a time scale of seconds. So in a football match we have 90 * 60 = 5400 seconds at least. Finally we have speed. Let’s go with 11 different speeds going from 0 m/s to 10m/s which is very fast.

So this gives the following number of options for a single player with or without the ball on an empty pitch. The formula is:

Options = position * direction * Speed * Timing * Ball

Which gives us:

6400 * 12 * 11 * 5400 * 2 = 9.123.840.000 options for a single player on an empty pitch. Over nine billion options is way to many options for people to handle conscious or unconsciously. That is why we need to decrease this number of options to make an interesting game that can actually be played by human beings.

Now if we add a single opponent, you can see the number of options for the first player drop only because there is 1 less position where he can be. Now our original player has the following number of options:

6399 * 12 * 11 * 5400 * 2 = 9.122.414.400 So now we see that by adding 1 extra inactive opponent, our original players has over a million option less. Now let’s add the other 20 players completely inactive with each player only taking up 1 space. The formula then becomes:

6379 * 12 * 11 * 5400 * 2 = 9.093.902.400 So that 29 million less options for our original player. Here you can see mathematically how adding more opponents actually decreases complexity. And this does not even take into account rules like off side where the opposing players can actively make large parts of the pitch inaccessible. Nor does it take into account all the seconds that the opposing team has the ball and the “with or without the ball” variable drops from 2 options to 1 option immediately limiting all available options for our original player for all those seconds by half.

With every second ticking away in a match, the number of options available to players decreases. One of the reasons why a formation helps players perform better is that a formation limits the number of positions where players can be, thus decreasing complexity.

## Complexity and space

So far we have considered the whole pitch as the boundaries of the system. But we can actually look at smaller parts of the pitch. What goes for the whole pitch, goes for every part of the pitch: the less opponents there are in any given space, the more options you get and the more complex the game becomes. And the more complex the game becomes, the more chances you get to score. So once you have divided the pitch into zones, it becomes important to make sure that your team has more players in the important zones than your opponent. The better the ratio is between your players and the opposing players, the more options your players have. Complexity rises, but so do the chances to score.

This only works of course if your players are able to handle the increased complexity. A well known phenomena in football is that if a player gets too much space and time, he starts to overthink the situation and blunder and lose the ball. The problem here is that the brain and the unconscious mind are suddenly confronted with more options than it can handle. The brain and the unconscious mind need consciousness to help out with the increase in complexity. Unfortunately, for some players their conscious mind is not trained to solve this level of complexity and a blunder happens.

If your team is not in possession of the ball, you want to close down the space of your opponent. The less space your opponent gets, the less options he has, the less complex the system becomes and the more predictable the behavior of your opponent becomes. Which, in the end, lowers his chance to score. And makes it easier to recover the ball.

Yet, the moment your team captures the ball, space works the other way around. Then you want less opponents in your space so you get more options. That is why Ten Hag, for instance, wants all players to spread out when the ball is lost, except for the few players who are needed to recover the ball. That way, once the ball is recovered, the team has way more options. The game has become a lot more complex for the opposing team and your team has an increased chance to score.

Then, once you run into a packed defense you want as many of your own team mates closeby and into the same space that the player with the ball occupies. This makes the ratio between your players and opposing player more even and increases complexity. Because if in the limited space where the ball is, the opposing team has more defenders than your team has attackers, these defenders diminish your options quickly, making your play more predictable and decreases your chances to score. The whole idea of parking the bus is to always have more defenders so to decrease complexity, make the game more predictable and decrease the chances of conceding a goal. At the same time, you don’t want your players to be so close together that they start to limit each others options. Players still need room to manoeuvre and run.

Here is an example of six attackers flooding a relative small space to increase complexity, making it harder to defend and easier to score:

Sheffield United’s winning goal today shows how their famous overlapping centre backs work: O’Connell pushes the team forward and creates numerical advantage on the left wing with Stevens and Freeman. Scored by Lundstram after the first shot, by Freeman, is stopped. pic.twitter.com/ldih9IPhGy

— Last Row (@lastrowview) August 18, 2019

Complexity also explains why sometimes it looks as if a team playing with ten players, because a player was sent off the pitch, is easier than playing with eleven. Sometimes, the manager even comments on this after the match by saying that he ought to play more often with ten players as it seems as if the team was playing better with ten players than eleven. In fact, this not only seems to be the case, but often this really is the case. Complexity explains it. The team has a lot less options with ten players against eleven. So the complexity has decreased for the team. If the team was struggling with the level of complexity when they were still with eleven players, it could well be that now with the decreased level of complexity they can manage it.

To be clear: for the opposing team, who still play with eleven players, the complexity has risen. This is why it is easier to score against a team of ten players than against a team of eleven players. But only if the players of this team are able to cope with the increase in complexity. Sometimes this is not the case.

## Complexity and risk

Ten Hag’s approach of attacking by making the space even smaller and adding more attackers into that small space, is one way to break through a defense by increasing complexity. Another example is to add one or more creative players to the lineup.

Why is a player called creative? Because he has more diversity in his play. He is less predictable than other players. The reason is that he has more variation. Adding a creative player to the team increases the complexity of the system. That is the reason why a creative player increases the chances to score.

But the more options you have, the more that can go wrong. Increasing complexity, also increases your chances to score. But it also increases the risks you take. For not only are there more ways in which things can go wrong, often the additional options creative players have, are also harder to execute. So the creative player needs more game intelligence and more technique than the average player. For all players, it is important to check to see whether they have a positive error ratio, but for creative players even more so. For a creative player might also increase the risk the team runs to an unacceptable high level.

Ten Hag’s strategy of spreading out on ball loss and concentrating attacks, does increase the chance of his team winning. It also increases the risks as the higher level of complexity might bite him. That is why you see him sometimes lose a game out of nowhere.

## The difference between complexity and difficulty

One can argue that scoring in a match without an opponent is very easy. And they would be correct. But the fact that scoring is easy with no opponent, doesn’t mean that it is not complex. For there is a difference between complexity and difficulty. Activities can be both complex and difficult. Or they can both be without much complexity and difficulty. Or they can be either complex and easy or having less complexity and yet be difficult to execute.

Complexity has to do with how many options you have. Not whether those options are easy or difficult to perform. In football there is a difference between decision making and executing decisions. The more complex the situation, the more options you have, the harder it becomes to make a decision. The less complex the situation is, the less options you have the easier it becomes to choose. In the example of a match against no opponent, even though most complex, it is easy to choose, because although there are so many options, a few options present themselves as most attractive because they are the easiest to perform. For instance, dribbling to the goal and score. But this is only the case if you think about the match as a single action. As soon as you understand that the players of your team have to enjoy themselves for 90 minutes, it becomes clear that they will entertain and probably do a lot of the other, less easy, options in that situation.

The difference between difficulty and complexity can also be seen in the following example:

Let’s do another experiment. This time taking a penalty. But before we take the penalty, we are going to strongly limit the number of options as to where to shoot by boarding up the goal with wooden planks. We will only leave a small hole in the middle of the goal just big enough to let the ball through. This is a real life example of a system with a very low complexity (as there is only one option to score), yet where the chosen action is very difficult to execute.

Now we can expand this example by creating a second hole in one of the corners of the goal. We make the second hole a bit bigger than the first hole so that the difficulty of the exercise stays the same. Although the difficulty is the same as in our previous experiment, the complexity has risen as now there are two options. And this proves that there is a difference between the difficulty of the execution of the decision and the complexity of the decision itself.

Of course we can continue to create more holes in the goal to increase complexity. Or we can remove all the planks and put a goalkeeper in the goal instead. Suddenly, a whole lot more options are available, complexity has risen, but it has also become easier to score. Some players will turn out to be better at handling the complexity of taking a penalty in the same way that other players will turn out to be better at executing the penalty. Ideally, you have a player in your team who is both good at dealing with the complexity of a penalty and at the execution of a penalty.

## Example of complexity in football

Here is another great tweet that explains complexity:

Espacio y Tiempo pic.twitter.com/PbmzOgEAOl

— Caño Football (@CanoFootball) September 5, 2019

What is so great about it is that it explains that if you give Messi space, he also gets time to think and he becomes very dangerous. But why does he become so dangerous if he has time to think? It is because he now has more options (more variation) and time to go through these options. So the complexity for Messi rises. Lesser players might be overwhelmed by this increase in complexity. But Messi is such a good player that he uses the increased complexity to become a lot more dangerous. So this is another great example of why more space and time increase complexity.