Modeling ball possession dynamics in the game of football

In this paper, we study interaction dynamics in the game of football–soccer in the context of ball possession intervals. To do so, we analyze a database comprising one season of the five major football leagues of Europe. Using this input, we developed a stochastic model based on three agents: two teammates and one defender. Despite its simplicity, the model is able to capture, in good approximation, the statistical behavior of possession times, pass lengths, and number of passes performed. In the last section, we show that the model's dynamics can be mapped into a Wiener process with drift and an absorbing barrier.

Figure 1

Relevant statistical patterns gathered from the data set Events in Ref. [18]. (a) Frequency by type of event. Blue bars, from the set of all the events. Red bars, only the events triggering a ball possession change (BPC). (b) The main plot shows the number of different players involved in a ball possession interval (BPI). The inset shows the number of different types of events in a BPI. $(*)$ The acronym OOTB stands for others on the ball.

Figure 2

Scheme summarizing the main parameters of the model (not to scale). Green circles represent the teammates and orange circle the defender. (a) We emphasize the parameters (i) $d$, the distance between the player with the ball and the defender, (ii) $d_x$, the distance between the player with the ball and the free player, and (iii) $a$, the action radius. (b) The circles placed at distance $R_1$ from the origin represent the initial condition in the dynamics. Distance $d_0$ is the initial distance between the three agents. Radius $R_2$ delimits the agents' moving area.

Figure 3

Relevant statistical observables found in the data set Events (DS) in Ref. [18], compared with the model outcomes (MO). For the results shown in the four panels, we have set the parameters of the model with the values $a=1$, $p=0.3$, $R_1=2.25$ and $R_2=16$. (a) The main plot shows the distribution of the possession time $T$, whereas the inset shows the distribution of the time differences between two consecutive events, $P\left(\mathrm{\Delta }t\right)$. (b) Distribution of the distance between two consecutive events segmented in the groups: (i) the whole set of events $P\left(\mathrm{\Delta }r\right)$, (ii) the passes tagging as sub–type simple pass $P(\mathrm{\Delta }r,X_1)$, (iii) the passes tagging with any other sub-type $P(\mathrm{\Delta }r,X_2)$, and (iv) all the passes $P(\mathrm{\Delta }r,X)$. Notice, the plot is in linear–log scale. (c) Distribution of the number of passes in the ball possession intervals, $P\left(N\right)$. (d) Mean value of the number of passes, as a function of the possession time. The blue dashed line indicates a linear fit $〈N〉={\omega }_{p}T$ performed on this region, with ${\omega }_p=0.19\pm 0.03(1/s)$.

Figure 4

Results of mapping the model to a Wiener process with drift and an absorbing barrier. (a) Distribution of steps $\delta$, segmented in all the data, $P\left(\delta \right)$, those steps given in the context of a simple persecution, $P(\delta ,S_1)$, and those steps in the context of a pass, $P(\delta ,S_2)$. (b) Nonlinear fit performed to distribution $P\left(T\right)$ (MO), using the expression $g\left(t\right)$ given by Eq. (2).