Storyboard

Using game videos, an estimation of the key parameters of the model can be made:

- average time between passes

- average distance advanced or retreated with the ball

- probability of kicking forward or backward

- probability of keeping the ball

- average distance thrown by the goalkeeper

>Model

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The game starts in the middle of the field, being the first movement of one of the two teams in a defined way (other players cannot be in the vicinity):

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

The game starts ($t=0$) in a defined position ($x=0$), with one team having the ball.

It is played on a court of length $L=125,m$, so in the selected coordinate system, the goals are in the positions $-L/2=-62.5\ ,m$ and $L/2=62.5,m$ respectively.

All movements are projected on the axis that links both arcs, so lateral movements are not modeled explicitly.

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The team A player may advance alone with each kick counted as a team move:

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

After kicking the ball there is a $p_A$ probability that he will maintain his control and succeed in kicking it again.

If it fails to control it, the opposing team B will be able to take control of it with probability $p_B$.

To simplify, it is assumed that the control probabilities are specific to the team and are assumed to be homogeneous among the players and constant during the event.

For the simulation we conclude:

The game can be modeled as sequences of shots that we will call moves.

Each move has a length that can assume values around a mean value $\Delta l$.

Each movement has an average duration $\Delta t$ that will be used as the basis for the simulation.

For simplicity it is assumed that the length of the move and the length of the time interval are the same for both teams and all players.

There is a probability $p$ that the player will be able to maintain control of the ball at each move. If they lose the ball, the opposing team will be able to make a move after which they will be similarly exposed to the possibility of losing the ball.

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A special play that should be considered is when the player makes a pass, that is, he kicks the ball looking for another player from the same team to take control:

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

For simplicity, it is assumed that the probability of maintaining control is the same value that was assumed for the team independent of the player but typical of the event.

For the simulation we conclude:

The pass is a movement between players of the same team, so it is equivalent to one more move.

Although the length of the pass can be greater than the distance traveled between two shots, it is assumed for simplicity that the length of a move is the average of the lengths of all moves including those made with passes.

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If the player finds that his progress is obstructed, he has the possibility of going back or making a pass to a teammate who is in his rear.

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

If the movements of a series of moves are observed, it is noted that in the center the shots are in all directions while in the extremes they are oriented either to avoid an own goal or to convert in the opposing goal:

For the simulation we conclude:

You must enter an angle $\theta$ with which the shots are launched, which have different probabilities depending on the position on the court.

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If it is assumed that the angle at which the shots are executed varies from covering 180 degrees in the own arc to 360 degrees in the center and then as it descends to 0 in the opposite arc:

movements along the axis between the arcs can be estimated:

In this case the path traveled in the projection can be calculated from the angle.

For the simulation we conclude:

According to the player\'s position on the field, a random $\theta$ angle must be generated in the defined range.

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A special shot is the shot to the archery:

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

Unlike normal shots, in which the same team is expected to control the ball at the end of the movement, in a shot to the archery control passes to the opposing team. Whether or not he converts depends on whether the team manages to control the ball and thus avoid the goal. Therefore, the goal will depend on the fact that the opposing team, with its control probability $p$, does not achieve it, which corresponds to the probability $1-p$.

For the simulation we conclude:

The shot to the archery itself is one more move that is exposed to being controlled by the opposing team.

The chance to convert is equal to the chance that the team with control chance $p$ fails, which is equal to $1-p$.

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The shot by the goalkeeper is special in that it can be such that it throws the ball to a particular player

From Chelsea 4-0 Manchester United, Premier League Replay (2022)

For the simulation we conclude:

The average distance with which the goalkeeper throws the ball is generally different from the distance of a move, so it is necessary to define an average distance $\Delta L$ with its respective deviation $\sigma_{\Delta L}$ .

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