The Relationship Between Player Density and the Nature of a Game

A Game Theoretical Analysis of Sports

When I was growing up in Rio de Janeiro, Brazil, I used to play soccer at school. It was wild. The school was in the middle of the city, so the field wasn’t large. The ball was often kicked off the field, ending up on a roof or in the busy street. The field was made of concrete. The injuries were often brutal. The concrete acted like a grater when it hit our legs and knees. I had some gnarly injuries back in those days.

A philosopher playing soccer (Monty Python)

These days, everyone is talking about soccer, thanks to the World Cup. In one of those conversations, my mom mentioned that soccer is kind of boring because it is too slow. But thinking back to the old days of playing soccer in Brazil, I noticed that, unlike the typical soccer matches we watch on TV, the game was always very fast-paced. It was insane. Every match was filled with non-stop action. We were always running, always taking the ball from one another, and every match was filled with goals. That got me thinking – what was different between the matches I played at school and the World Cup matches we watch on TV? I quickly applied systems theory and game theory to the problem, which led to an interesting realization. It all comes down to the density of players on the field.

Soccer is typically played on very large fields by teams of 11 players, including a goalkeeper. A typical soccer field is between 100 and 130 yards long and 50 and 100 yards wide. This means there is always a significant distance between players on the field. It also takes a while to cross the field from one side to the other. The large distance between players, coupled with the low number of players relative to the field size, slows the game. As the game slows down, it becomes more strategic. In this official version of the game, players occupy more clearly defined positions that correspond to their locations on the field. Some players stay close to their team’s goal, defending it, while others stay close to the other team’s goal, attacking it. There are also players in the middle of the field, connecting defenders and attackers. These positions also come with different skill sets. Thus, a player can be good at one position but not at another. The other team is also organized in that way. Given the competition between teams for the ball, their positions on the field, and the rules of the game, it follows that on a large field with low player density, the game becomes slower and more strategic.

If you reduce the field size while keeping the number of players constant, you increase the player density on the field. In a small field, players don’t need to move as much to reach each other and the other team’s goal. That results in more interactions between the players. The game also speeds up, with players moving faster through the field and the ball changing possession more quickly. Finally, when the field is small enough, positions in the field don’t matter as much. Any player can try to score a goal at any time from anywhere on the field, something that is impossible in a large field. Rules can also change. Offset doesn’t make much sense anymore, since the field is so small. In a more enclosed field, like the one I used to play, there are no sidelines, so the ball just hits the wall and bounces back into the field. There are no throw-ins to slow down the game. The players and the ball are constantly moving and interacting. There is no time to think; players must just move. All of these factors make the game more tactical than strategic.

This pattern is observed in similar sports. Take hockey or volleyball, for instance. Both sports are played on much smaller fields than soccer and are much faster and livelier matches. Their fields are much smaller, and the player density is much higher. Players also score far more often. On the other hand, take football and baseball. Both are played on much larger fields, with much lower player density. They are much slower games with more cumbersome rules that slow them down. They are also far more strategic than hockey or volleyball.

In a nutshell, player density significantly shapes the nature of a game. The structure of games is influenced not only by their rules but also by their player density. This affects the speed of interactions in the game as well as the game's overall nature. All games have both tactics and strategy. However, games with lower player density are more strategic, while those with higher player density are more tactical. Strategy involves long-term planning designed to achieve a goal, while tactics are the short-term actions taken to execute a plan.

Players in the field have to decide how to score a goal. Strategy is involved in the specific configuration of player distribution on the field, their “formation”. You have a specific distribution of players, and the other team has its own. The strategy of soccer is about figuring out the best distribution for your team and against a given opponent. You need to know their distribution and be able to adapt to it. The execution is, obviously, important, so tactics are important as well. However, the tactics are constrained by the strategy, and the strategy exists only because the field is large enough and player density is low enough to necessitate a strategy to score goals.

It is worth noting that these patterns are universal. When a system becomes smaller without changing the number of its constituents, interactions among them speed up. Conversely, if the size of the system remains constant but more constituents are introduced, the resulting higher density accelerates interactions among them. If the change is significant enough, it alters the nature of the system. A star is a great example of that. Stars are composed of atoms that bounce off one another, producing energy. When a star absorbs more matter, the density of atoms increases, leading to faster interactions between atoms. If enough matter is introduced into the star, its nature changes according to astrophysical rules. The only reason there are no strategies or tactics inside stars is that, unlike people, atoms are not agents. There are no games without agents to play them. However, the patterns of interaction are generally governed by the underlying rules of systems.

Soccer, as it is officially played, is a strategic sport. Winning a game of soccer requires anticipating the other team’s strategy and adapting to it. The large fields slow the game down, potentially making it boring to watch. However, if the field were significantly smaller, the game would speed up, rely more on tactics than strategy, and probably be more fun to watch. But that would change the nature of soccer. It is not clear whether that would still be soccer, but I used to call the game I played as a child “soccer”. Either way, maybe FIFA should take some notes.