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Athens Journal of Sports - Volume 10, Issue 4, December 2023 – Pages 215-234
https://doi.org/10.30958/ajspo.10-4-2 doi=10.30958/ajspo.10-4-2
A Mathematical Analysis of Team Impact and Individual
Player Contribution in Football
By Jeffrey Leela
∗
, Donna M. G. Comissiong
±
& Karim Rahaman
°
In this paper, we present an important application of the Hungarian Method - a
well-known combinatorial optimization tool for solving assignment problems.
For our purposes, we consider the assignment of players to specific roles in a
football team. It involves the broad classification of team players as defensive,
midfield or attacking, while assigning the main roles associated with each of
these positions. This provides insight on specific role of each individual player,
thereby facilitating an optimal team selection. To illustrate this method, we
utilize the average player statistics per game for two teams from the 2016/2017
Premier League Season. In addition, a team rating index is created by identifying
six sub-indices. The first is called team contributions - which includes set piece
goals, percentage tackles won, percentage take-ons won, percentage aerial
duels won, number of interceptions, number of blocked shots, number of
clearances, number of red and yellow cards. To visualize the method, a multiple
correlation is carried out on team data for the 2016/2017 Premier League
season to generate a correlation coefficient for each contribution. The resulting
team index can be a useful tool for measuring the overall strengths of competing
teams in a football league.
Keywords: Hungarian method, football, team rating index, multiple correlations,
team comparisons
Introduction
It is a well-known fact that the most successful teams are the ones that are
best balanced, not necessarily those comprised of the best collection of available
players. Nevertheless, each player’s individual contribution is vital for the overall
team performance, and coaches are continually seeking the most effective
techniques for identifying the most outstanding players. Modern-day football
scouts can make use of data-driven analysis techniques to assess any player’s
potential based on the available performance metrics. After recruiting the players
with the best ratings, it is then up to the coaching staff to conduct appropriate
training sessions to get the players to work together, harnessing each individual
player’s strengths to optimize overall team performance.
One of the most important tasks of a football coach is team selection,
according to the match being played, and after careful consideration of the
∗Senior Lecturer, Department of Mathematics, The College of Science, Technology and Applied
Arts of Trinidad and Tobago, Trinidad, West Indies.
±Senior Lecturer, Department of Mathematics and Statistics, The University of The West Indies, St.
Augustine Campus, Trinidad, West Indies.
°Senior Lecturer, Department of Mathematics and Statistics, The University of The West Indies, St.
Augustine Campus, Trinidad, West Indies.
Vol. 10, No.4
Leela et al.: A Mathematical Analysis of Team Impact and Individual…
216
opposing team’s strengths and weaknesses. While is not necessarily true that a
collection of players with the best individual performance ratings would be the
optimal choice for the selected team, once these players have trained together and
have a well-defined game plan, it is not unreasonable to expect a favourable match
outcome. The difficulty lies with the process of team selection when the available
players have similar attributes. In such cases, it would be beneficial for coaches to
have a scientific method to distinguish between closely matched players with
similar abilities. Given the large sums of money on offer for winning professional
football leagues and national team titles, the ability to select the most suitable team
players has become an indispensable skill (Qadar et al. 2017).
The Hungarian method can be employed for the effective solution of
assignment problems, where a set of tasks must be assigned to workers who each
have a different level of ability. The problem is solved by creating the cost matrix
associated with each worker-task pair, and consequently finding the optimal
assignment of workers to tasks through a series of iterative steps. The objective is
to minimize the total cost or to maximize the total benefit associated with the
completion of the assigned jobs. As the Hungarian method guarantees an
assignment solution that is both feasible and optimal, it can conceivably be
employed to determine the optimal team selection for any team sport. The method
was successfully applied by Britz and Maltitz for the optimal selection of baseball
players for the most effective team (Britz and Maltitz 2010). After assessing a
group of novice baseball players to determine their abilities in key practical
aspects of the game, they successfully employed the Hungarian method to
determine the optimal team, according to these metrics. This same approach could
conceivably be adapted for team selection in football, utilizing the available player
performance metrics freely available on online football data repositories.
In this paper, we utilize these player performance metrics – available for free
download from Whoscored.com – to create an efficiency matrix for key players in
a football team. We do this by classifying players according to their specific role
on the team, and extracting the relevant statistical data associated with the jobs
typically allocated to defenders, midfielders and strikers. Subsequently, we apply
the Hungarian algorithm to the efficiency matrix to determine the maximal
defensive, midfield and striking scores for the team. This facilitates an unbiased
comparison of competing teams in a professional football league using
summarised player statistics obtained from a recently completed season. Given
that large sums of money are spent each year on recruiting new players, it would
be very helpful to have another scientific tool to analyse immediate past team
results to effectively identify problem areas where player recruitment might be
opportune.
Next, we present a general method for the overall rating of teams in a
professional football league, using the Premier League to illustrate this objective.
To formulate our team rating system, we establish a set of criteria which
characterizes all-round team play. Inspired by the player ranking methodology
presented by McHale et al., we introduce an appropriate number of sub-indices,
the first being called “team contributions” which is sub-divided into set piece
goals, duels won percentage, defensive actions and discipline (McHale et al.
Athens Journal of Sports
December 2023
217
2012). The required data for each registered team for the 2016/2017 Premier
League season was sourced from Squawka.com (Squawka 2017). A multiple
correlation analysis is performed with points achieved by each team as the
reference variable, with the other variables being set piece goals, tackles, take-on,
aerial duels won, interceptions, blocked shots, clearances, red and yellow cards.
As football fans around the world will attest, the final result of a match does
not often represent the actual performance of a football team. Our proposed team
index is a single score that effectively rates the collective contributions of all team
players. While there are several predictive tools that are available for use in team
football, our analysis will provide an avenue for evaluating the overall team
performance after the season has ended. A quick comparison with the overall team
standings at the end of the playing season can easily demonstrate the effectiveness
of the method, lending credibility to its usefulness for coaching staff when
planning for future seasons.
Literature Review
The mathematical foundation for the Hungarian algorithm was established by
the Hungarian mathematicians Konig (1913) and Egevary (1931). Harold Kuhn
later devised a computational algorithm that efficiently employs the Hungarian
method for the solution of an assignment problem (Kuhn 1955). The algorithm
was studied independently by James Munkres in 1957, and for that reason, it is
sometimes referred to as the Kuhn–Munkres algorithm or the Munkres assignment
algorithm (Munkres 1957). The method reduces the associated cost matrix in such
a manner that at least one zero in each row and column will be obtained. The
positions of these zeros in the matrix are representative of the optimal assignment
solution, thus facilitating the calculation of the minimal opportunity cost.
Britz and Maltitz utilized the Hungarian algorithm for team selection in
baseball, by assigning the most effective player to respective positions on the field
(Britz and Maltitz 2010). They considered different weighted combinations of
player roles on a baseball field, while considering the overall balance that must be
achieved between offensive and defensive plays. They then tested their proposal
on a group of novice baseball players by conducting skill tests to determine the
relevant ratings for the associated efficiency matrix. They then employed the
Hungarian algorithm to identify the optimal team. To the best of our knowledge,
this methodology has not yet been adopted for team selection purposes in football.
As explained by McHale et al., “performance assessment is a fundamental
tool for quantitative analysts and operational researchers” (McHale et al. 2012).
Rating systems are often utilized to measure team or player performance, and there
are well-established rating systems for ranking opposing teams in competitive
sports competitions. In individual sports such as tennis, it is relatively
straightforward to analyse recent results of player competitions to generate an
ordered list of the top ranked players. As these official rankings are often used to
seed players in a tournament, this can also affect the overall outcome of the
tournament, as top seeded players are effectively guaranteed an easier route to the
Vol. 10, No.4
Leela et al.: A Mathematical Analysis of Team Impact and Individual…
218
final rounds of matches. It is true however that there are limitations to any ranking
system, and absolute trust cannot be placed on rating systems that rely only on past
player performances. Even the official rankings provided by the well-established
Association of Tennis Professionals (ATP) might prove somewhat deceptive for
sports enthusiasts placing bets on the top ranked tennis players (McHale and
Morton 2011).
Tennis is not the only sport to have used officials’ rankings to predict future
performance. Forrest and McHale found that for men’s professional golf,
increased forecasting power can be achieved by incorporating up-to-date results
with an established forecasting model which utilizes world rankings as a predictor
(Forrest and McHale 2007). In a similar study for football, McHale and Davies
determined that recent match results of international teams can add much value to
the forecasting model (McHale and Davies 2007). Thus, the evidence from tennis,
golf and football suggests that although official rankings of players and teams are
useful as predictors, they do not determine match outcomes with absolute
certainty. Reliable team ratings are required for the calculation of betting odds, and
substantial funds are generated when sports fans place bets on their preferred
teams. The availability of methods for the evaluation of team performance is
therefore of great interest not only to players and coaching staff, but also to the
wider community of sports enthusiasts.
Methodology
Individual Player Contribution
In assignment problems, the main objective is the allocation of jobs to an
equal number of persons at a minimum cost for maximal profit. Let us suppose
that there are ‘n’ jobs to be performed and ‘n’ persons available to take these jobs.
We assume that each person can complete an assigned job in a specified time with
a varying level of efficiency. Let
ij
c
be the cost associated with the ith person being
assigned to the jth job. Our goal is to determine the optimal job assignment such
that the total cost for performing all the jobs is minimized. Typical examples of
assignment problems include the allocation of machines to jobs, classes to
classrooms, players to a team, etc.
Basic Mathematical Formulation
Cost matrix:
ij
c
= c
11
c
12
…………………..c
n1
21
c
c
21
………………….c
n2
… … ………………….
c
n1
c
n2
…………………..c
nn
We wish to minimize cost: z =
∑∑
==
n
1i
n
1j ijijxc
i = 1, 2,…, n ; j = 1, 2,…, n.
Athens Journal of Sports
December 2023
219
subject to the conditions
x
ij
=
otherwise 0
job j toassigned isperson i if 1
thth
n)2,..., 1,iperson i by the done is job (one 1x
th
n
1j ij
==
∑
=
n)2,..., 1,i job j theassigned be shouldperson one(only 1x
th
n
1i ij
==
∑
=
where
ij
x
denotes the j
th
job to be assigned to the i
th
person.
The Hungarian Algorithm (Britz and Maltitz 2010)
The position that a player occupies on the field defines the role and
responsibility of that particular player. There are three main positions for outfield
players in a football team: defender, midfielder or striker. Players may be asked to
perform multiple tasks/jobs in accordance with the team formation/tactical
directives provided by the coaching staff. These jobs include passing, tackling,
blocking, intercepting, clearing, shooting, assisting, and dribbling. Most football
players are better at mastering one or two of these jobs, although there are a few
exceptional players who exhibit extraordinary levels of talent and can therefore
perform multiple functions with equally high levels of competence. In general,
regardless of the position that they occupy, players must be able to perform all
these jobs effectively - since football is a team sport, and successful teams are
comprised of players who can adapt quickly to changing situations on the pitch.
Vol. 10, No.4
Leela et al.: A Mathematical Analysis of Team Impact and Individual…
220
To illustrate the method, we utilize the average player statistics per game for
two teams from the 2016/2017 Premier League Season (Whoscored 2017). We
select the team that placed first that year: Chelsea, and the team that placed sixth:
Manchester United. Our main objective is to investigate the roles performed by the
players from each team, and in so doing, to provide reasons for the gulf in class
between these two teams. This type of critical analysis can help the coaching staff
to identify what is working well for their team, and what needs to be improved.
We begin by classifying the players on each team according to their main
roles – as defender, midfielder or striker. Defenders are given five major jobs
while midfielders are given seven and strikers four. As midfielders must perform
both defensive and offensive duties, there is some overlap in the tasks to be
performed by defenders and midfielders as well as by midfielders and strikers. We
use the available data to assign the players to jobs, noting that whenever there are
more players than jobs, the resulting matrix is not square. As the Hungarian
Algorithm requires a square matrix, in such cases, “dummy jobs” must be created
to facilitate the analysis. Although not all the players will be given a legitimate job
as a result, the analysis will still allow us to identify the most efficient combination
of players on the team to perform all the tasks outlined. Our objective is to
maximize each team’s defensive and offensive statistics, based on the available
data. This will allow important comparisons to be made between the two teams.
Our results will provide reasonable justifications for the gap in points scored
between the teams and for the overall performance of each team as a unit.
We will illustrate the method by considering the defensive statistics for
Manchester United. The nine defenders used for the majority of the 2016/2017
Premier League season by Manchester United are listed in Table 1 with their
associated averages for the five defensive jobs considered crucial for their
position. Note that each number indicates the average for that particular job per
game, and that passing data is based on quoted pass percentages. For example,
Smalling has an 89% successful passing rate.
Table 1. Defenders for Manchester United
Smalling Blind Valencia Rojo Young Bailly Shaw Darmian Jones
Tackling 0.7 2.0 2.4 1.4 1.5 2.4 1.1 2.4 2.1
Clearing 6.9 4.7 2.1 6.8 2.1 5.0 2.5 4.1 7.6
Blocking 0.5 0.4 0.3 0.3 0.2 0.9 0.1 0.1 0.8
Intercepting 0.7 1.9 1.5 1.6 1.3 2.5 1.1 2.3 1.6
Passing
0.89
0.86
0.86
0.86
0.83
0.86
0.86
0.81
0.89
Our problem is to maximize the defensive statistics for the team - by
identifying the combination of five selected players that results in the maximum
overall defensive score for the team with respect to the five tasks identified:
talking, clearing, blocking, intercepting, and passing. Now, to turn this into a
maximization type problem for the Hungarian algorithm we must first develop the
effective matrix. To do this, we must first subtract the largest entry (7.6) from each
other entry of the matrix. The resulting matrix is shown in Table 2.
Athens Journal of Sports
December 2023
221
Table 2. Subtract the Smallest Entry from Each Row from all other Entries in that
Row
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling 6.9 5.6 5.2 6.2 6.1 5.2 6.5 5.2 5.5
Clearing
0.7
2.9
5.5
0.8
5.5
2.6
5.1
3.5
0
Blocking
7.1
7.2
7.3
7.3
7.4
6.7
7.5
7.5
6.8
Intercepting
6.9
5.7
6.1
6.0
6.3
5.1
6.5
5.3
6.0
Passing
6.71
6.74
6.74
6.74
6.77
6.74
6.74
6.79
6.71
Next, we add dummy jobs to make the number of rows to equal the number of
columns. The dummy jobs are denoted as F, G, H and I, as illustrated in Table 3.
Table 3. Effective Matrix – After Addition of Dummy Jobs F, G, H and I
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling 6.9 5.6 5.2 6.2 6.1 5.2 6.5 5.2 5.5
Clearing
0.7
2.9
5.5
0.8
5.5
2.6
5.1
3.5
0
Blocking
7.1
7.2
7.3
7.3
7.4
6.7
7.5
7.5
6.8
Intercepting 6.9 5.7 6.1 6.0 6.3 5.1 6.5 5.3 6.0
Passing
6.71
6.74
6.74
6.74
6.77
6.74
6.74
6.79
6.71
F
0
0
0
0
0
0
0
0
0
G 0 0 0 0 0 0 0 0 0
H 0 0 0 0 0 0 0 0 0
I
0
0
0
0
0
0
0
0
0
We can now proceed with the steps listed in the Hungarian algorithm.
Subtract the minimum element in each row from each element in that row. As
there are zeros in every column, there is no need to subtract the minimum element
from each column from all elements in that column. The result is shown in Table 4.
Table 4. Modified Matrix – After Subtraction of Minimum Element from all Rows
Smalling Blind Valencia Rojo Young Bailly Shaw Darmian Jones
Tackling
1.7
0.4
0
1
0.9
0
1.3
0
0.3
Clearing
0.7
2.9
5.5
0.8
5.5
2.6
5.1
3.5
0
Blocking
0.4
0.5
0.6
0.6
0.7
0
0.8
0.8
0.1
Intercepting
1.8
0.6
1
0.9
1.2
0
1.4
0.2
0.9
Passing
0
0.03
0.03
0.03
0.06
0.03
0.03
0.08
0
F
0
0
0
0
0
0
0
0
0
G
0
0
0
0
0
0
0
0
0
H
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
As there are zeros in every column, there is no need to subtract the minimum
element from each column. We must now cover all the zeros with the minimum
number of horizontal and vertical lines. This yields eight lines, as shown in Table 5.
Table 5. Cover all Zeros with Minimum Number (8) of Horizontal/Vertical Lines
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling
1.7
0.4
0
1.0
0.9
0
1.3
0
0.3
Vol. 10, No.4
Leela et al.: A Mathematical Analysis of Team Impact and Individual…
222
Clearing
0.7
2.9
5.5
0.8
5.5
2.6
5.1
3.5
0
Blocking
0.4
0.5
0.6
0.6
0.7
0
0.8
0.8
0.1
Intercepting
1.8
0.6
1
0.9
1.2
0
1.4
0.2
0.9
Passing
0
0.03
0.03
0.03
0.06
0.03
0.03
0.08
0
F
0
0
0
0
0
0
0
0
0
G
0
0
0
0
0
0
0
0
0
H
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
As the order of the matrix is nine, the optimal assignment cannot be made.
We proceed by subtracting the minimum uncovered element from all uncovered
elements and add this minimum uncovered element to the covered elements at the
line intersections only. From Table 5, we see that the minimum uncovered element
is 0.03. To cover all the zeros with the minimum number of horizontal and vertical
lines in the resulting matrix, we will again require eight lines (as shown in Table
6), so once again, the optimal assignment cannot be made.
Table 6. Zeros Covered with the Minimum Number (8) of Horizontal/Vertical
Lines
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling
1.73
0.4
0
1.0
0.9
0.03
1.3
0
0.33
Clearing
0.7
2.87
5.47
0.77
5.47
2.6
5.07
3.47
0
Blocking
0.4
0.47
0.57
0.57
0.67
0
0.77
0.77
0.1
Intercepting
1.8
0.57
0.97
0.87
1.17
0
1.37
0.17
0.9
Passing
0
0
0
0
0.03
0.03
0
0.05
0
F
0.03
0
0
0
0
0.03
0
0
0.03
G
0.03
0
0
0
0
0.03
0
0
0.03
H
0.03
0
0
0
0
0.03
0
0
0.03
I
0.03
0
0
0
0
0.03
0
0
0.03
We must repeat the steps of the Hungarian algorithm. The smallest uncovered
number is 0.17, so we subtract 0.17 from all uncovered numbers, and we add 0.17
to the covered numbers that are located in any position where two lines intersect.
Nine lines can be used to cover all zeros in the resulting matrix, as shown in Table
7. The optimal assignment can now be determined.
Athens Journal of Sports
December 2023
223
Table 7. Zeros Covered with the Minimum Number (9) of Horizontal/ Vertical
Lines
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling
1.73
0.4
0
1.0
0.9
0.2
1.3
0
0.5
Clearing
0.53
2.7
5.3
0.6
5.3
2.6
4.9
3.3
0
Blocking
0.23
0.3
0.4
0.4
0.5
0
0.6
0.6
0.1
Intercepting
1.63
0.4
0.8
0.7
1
0
1.2
0
0.9
Passing
0
0
0
0
0.03
0.2
0
0.05
0.17
F
0.03
0
0
0
0
0.2
0
0
0.2
G
0.03
0
0
0
0
0.2
0
0
0.2
H
0.03
0
0
0
0
0.2
0
0
0.2
I
0.03
0
0
0
0
0.2
0
0
0.2
Zeros are then eliminated to leave one zero in each row and column, thus
ensuring that each player is assigned to one job. The resulting matrix is presented
in Table 8, where the symbol
⊗
indicates an eliminated zero while (0) indicates
the assigned player in the respective column with the corresponding job in the
respective row.
Table 8. Matrix Displaying the Optimal Assignment Solution
Smalling
Blind
Valencia
Rojo
Young
Bailly
Shaw
Darmian
Jones
Tackling
(0)
⊗
Clearing
(0)
Blocking
(0)
Intercepting
⊗
(0)
Passing
(0)
⊗
⊗
⊗
⊗
F
(0)
⊗
⊗
⊗
⊗
⊗
G
⊗
⊗
(0)
⊗
⊗
⊗
H
⊗
⊗
⊗
(0)
⊗
⊗
I
⊗
⊗
⊗
⊗
(0)
⊗
For verification purposes, we utilized the MATLAB Hungarian Algorithm for
linear assignment problems (V2.3) developed by Yi Cao (Yi 2023). Deployment
of the program “munkres.m” with the effective matrix (see Table 3) yielded the
same result displayed in Table 8, which effectively confirms our manual
calculations. The optimal team defensive score is subsequently calculated by
adding the original performance scores by the players selected for the optimal
solution (refer to Table 1). This type of analysis facilitates a comparison of
defensive systems employed by different teams in the Premier League, with the
highest overall defensive team score expected to correspond with the team with
the most effective defensive records.
Vol. 10, No.4
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224
A similar analysis can be carried out for midfield players and strikers, for
their respective jobs. For brevity, our calculations for the two football teams under
consideration will be summarized in the results section.
Analysis of Team Impact
In formulating a system for rating, we must first establish a number of criteria
which constitutes the all-round play of each registered team in the league under
consideration. The necessary data for each team for the 2016/2017 Premier League
Season was collected from Squawka (Squawka 2017). A multiple correlation
analysis was then carried out with ‘number of points obtained’ as the reference
variable with the other variables being ‘number of set piece goals’, ‘tackles
percentage won’, ‘take-ons percentage won’, ‘aerial duels percentage won’,
‘number of interceptions’, ‘number of blocked shots’, ‘number of clearances’,
‘number of red cards’ and ‘number of yellow cards’. The results are presented in
Table 9.
Table 9. Multiple Correlation Analysis Results
Multiple Correlation Analysis
Correlation Coefficients
# Set Piece Goals & # Points
0.4844
Tackles % Won & # Points
0.4239
Take-ons % Won & # Points
0.1644
Aerial Duels % Won & # Points
0.0333
# Interceptions & # Points
0.4651
# Blocked Shots & # Points
0.7879
# Clearances & # Points
0.6562
# Red Cards & # Points
0.2820
# Yellow Cards & # Points
0.2411
As positive correlation coefficients are indicative of a relationship between
two variables that tends to move in the same direction, our results indicate that the
more set piece goals scored by a team, the more points will be obtained. This is
also the case for tackles percentage won. We note that take-ons percentage won
and points are positively correlated but only distantly so, while aerial duels
percentage won and points have almost no correlation. This indicates that there
may be other underlying factors that we have not considered for those two
variables.
As expected, defensive actions and points are all negatively correlated, i.e.,
more of these actions is indicative of fewer points attained by the team. We note
that teams with a larger number of interceptions, blocked shots and clearances are
constantly under attack. As a result, those teams will be defending frantically to
stay in the game, with much less focus on offensive play. Discipline also impacts
Athens Journal of Sports
December 2023
225
on the team, so it is no surprise that red/ yellow cards and points are negatively
correlated.
We are now ready to establish the first sub-index of the rating system by
multiplying the number of actions of each team by the correlation coefficient
obtained for that action and then summing these products. As the number of set
piece goals, tackles won, take-ons won and aerial duels won will be significantly
less than the number of defensive actions (namely blocked shots, clearances and
interceptions), we expect that the associated index will be negative. In general,
team data for the number of set piece goals and the percentages of tackles, take-
ons and aerial duels won is in the tens, while the data corresponding to defensive
actions is markedly higher, often measuring in the hundreds or thousands. To
compensate for this imbalance, we multiply set piece goals by one hundred - since
the number of goals scored decides the match outcome and the number of points
awarded to the team. The percentages associated with tackles, take-ons and aerial
duels won will also be multiplied by one hundred. This will allow for a better
balance in terms of the tabulated offensive and defensive actions of each team.
Sub-Index 1
Team Contributions Index
=100[0.4844()+ 0 4239()+ 0 1644()+ 0 0333()]
04651() 07879()06562()0
282()02411()
where = number of set piece goals, = tackles % won, = take-ons %
won, = aerial duels % won, = number of interceptions, = number
of blocked shots, = number of clearances, = number of red cards,
= number of yellow cards.
Sub-Index 2
Goal Difference Index
This sub-index awards points to a team based on net goals. The specific
number of points awarded has been calculated by converting goals into points.
Over the 2016/2017 Premier League Season, there was a total of 1064 goals
scored, and 1056 points won. Therefore, we can estimate how many points one
goal is worth as
99250
1064
1056 ⋅=
points for each goal. This means that on this
index, a team receives 0.9925 points for each goal the team scores. The points
awarded to a team for goal difference is simply points per goal multiplied by a
team’s goal difference scaled by a factor of ten - to keep in line with the weight of
the first sub-index, as well as not to outweigh it.
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99250difference goal10I i2 ⋅××=
where = 1,2,3, ,20 denotes the each of the 20 teams in the Premier League
Season 2016/2017.
Sub-Index 3
Assists Index
An assist is defined as a pass which leads to a goal. Therefore, from our
previous estimate a goal is worth
99250⋅
points. We can place an assist on this
same scale. Hence, each assist by a team is worth
9925
0⋅
points. The points
awarded for the assist for each team is simply the number of assists multiplied by
the points for each assist. As for the previous sub-index, we scale by a factor of ten
to get:
=10 × assists× 0.9925
where = 1,2,3, ,20 denotes the each of the 20 teams in the Premier League
Season 2016/2017.
Sub-Index 4
Key Pass Index
A key pass is defined as a pass that creates a goal scoring opportunity. At
times, a key pass leads to an assist. The total chances created by each team is a
combination of the key passes and assists of each team. From the 2016/2017
Premier League Season Data there were a total of 7067 chances created. 717 of
these chances created were assists, therefore:
717
7067 ×100 =10 146%
From this analysis we can conclude that approximately 10% of the chances
created resulted in goals. This leads to approximately 90% of the total chances
created to be classified as key passes. Therefore, a chance created is nine times
more likely to not result in a goal as to result in a goal. The points awarded per
assist are
99250⋅
. As a result, the points awarded per key pass should be
9
99250⋅
which is close to one ninth of the value of an assist, i.e.,
1103
0⋅
. As
before, we scale by a factor of ten to obtain
=10 ×key passes× 0 1103
Athens Journal of Sports
December 2023
227
where = 1,2,3, ,20 denotes the each of the 20 teams in the Premier League
Season 2016/2017.
Sub Index 5
Work Rate Index
The seasonal points obtained per team based on distance covered which again
is scaled by a factor of ten (10).
Work rate:
i
20
1i
i
i
5distance
10 points covereddistance
I∑
=
××
=
where = 1,2,3, ,20 represents the each of the 20 teams in the Premier League
Season 2016/2017.
Work rate is a measure that contributes significantly less than the other sub-
indices. This is mainly because players in general tend to run more and cover more
distance when they are not in possession of the soccer ball. This could translate to
being under pressure from opposing teams. Hence, absorbing such pressure takes a
high level of concentration and should be merited. In terms of team rating this
would not place a team at the summit by any means. However, it could separate
teams with fine margins in ratings.
The final index is calculated by taking the sum of the five sub-indices
calculated previously:
The Final Index
54
321
III
II ++
++=
.
Note that some of the ideas in creating this index were utilised and modified
from (McHale et al., 2012).
Results
a. Hungarian Method Results:
The optimal defensive assignment (jobs → player) of Manchester United in
the 2016/2017 Premier League season was as follows:
Passing → Smalling; Tackling → Valencia; Blocking → Bailly;
Intercepting →Darmian; Clearing → Jones.
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To determine the maximum defensive assignment score for Manchester
United, we combine the initial average data for the specific job that is assigned to
each of these five defenders as follows:
0.89 + 2.4 + 0.9 + 2.3 + 7.6 = 14.09
The same analysis can be carried out for the midfielders and strikers from
Manchester United, resulting in the following assignments.
Midfield assignment for Manchester United:
Passing → Lingard; Shots per Game → Pogba; Through Balls →Mkhitarian;
Key Passes → Mata; Tackling →Fellaini; Assists → Herrera; Intercepting →
Carrick.
The associated maximum midfield assignment for Manchester United is
calculated to give
0.88 + 3.1 + 0.2 + 1.8 + 2 + 6 + 1.9 = 15.88
Striker assignment for Manchester United:
Assists → Martial; Successful Dribbles → Rashford;
Shots per Game → Ibrahimovic; Fouled per Game→ Rooney.
The maximum assignment for the strikers of Manchester United tallies to:
6 + 1.3 + 4.1 + 0.5 = 11.9
The same analysis on the Chelsea team for defense is as follows:
Intercepting → Azpilicueta; Blocking → Cahill; Clearing → Luiz;
Passing → Terry; Tackling → Aké.
The maximum assignment for defence in the Chelsea team is therefore:
1.9 + 0.5 + 5.3 + 0.92 + 2 = 10.62
We now apply the analysis to the Chelsea midfielders. The optimal
assignment is as follows:
Shots per Game → Hazard; Assists → Fabregas; Through Balls → Willian;
Intercepting → Matic; Key Passes → Oscar; Tackling → Kanté; Passing →
Loftus-Cheek.
The maximum midfield assignment score for Chelsea is therefore:
Athens Journal of Sports
December 2023
229
2.1 + 12 + 0.1 + 1.4 + 1.7 + 3.6 + 0.84 =21.74
The optimal assignment for the strikers of Chelsea produces the following:
Assists → Pedro; Successful Dribbles → Hazard;
Shots per Game → Costa; Fouled per Game→ Moses.
The maximum striker assignment for Chelsea calculates is therefore given as:
3.2 + 8 + 3.9 + 0.9 = 16.
b. Performance Rating Results:
Figure 1 presents a scatter diagram displaying the multiple correlations for
each variable. These are used in the calculation of the first sub-index ‘team
contributions’. The four remaining sub-indices are ‘goal difference’, ‘assist’, ‘key
pass’, ‘work rate’. The five calculated sub-indices are summarized in Tables 10
and 11.
Figure 1. Multiple Correlations with Reference Variable ‘Points’
Table 10. Sub-indices 1 & 2 - Team Contributions & Goal Difference
Team Score (Team Contributions
Index) Score
(Goal Difference Index)
Chelsea
153258⋅
10
516⋅
West Bromwich Albion
622694⋅
4079⋅−
Tottenham Hotspur
472899⋅
50595⋅
Swansea City
90.2665
13248⋅−
Liverpool
112944⋅
30357 ⋅
West Ham United
962577⋅
73168⋅−
Bournemouth
692629⋅
10119⋅−
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0246810 12
Points
S.P.Goals
Tackles
Take Ons
Aerial
Interceptions
Blocked Shots
Clearances
Red
Yellow
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Manchester City
412735⋅
93406⋅
Burnley
87
2214⋅
80158⋅−
Hull City
962431⋅
78426 ⋅−
Arsenal
942741⋅
53327 ⋅
Crystal Palace
152478⋅
03129 ⋅−
Everton
692518⋅
65178⋅
Watford
542196⋅
90227 ⋅−
Stoke City
602424⋅
88148⋅−
Leicester City
462123⋅
88148⋅−
Manchester United
512058⋅
13248⋅
Southampton
402404⋅
4869⋅−
Sunderland
072073⋅
00397 ⋅
−
Middlesbrough
022099⋅
05258⋅
−
Table 11. Sub-Indices 3, 4, 5 - Assist, Key Pass and Work Rate
Team
Score
(Assist Index)
Score
(Key Pass Index)
Score
(Work Rate Index)
Chelsea
80555⋅
17430⋅
3547⋅
West Bromwich Albion
45337 ⋅
85
276⋅
1823⋅
Tottenham Hotspur
58585⋅
84
490⋅
1844 ⋅
Swansea City
60317 ⋅
47
283⋅
6520⋅
Liverpool
88545⋅
42
486⋅
99639⋅
West Ham United
83287⋅
17355⋅
1822 ⋅
Bournemouth
30357 ⋅
90330⋅
4823⋅
Manchester City
10516⋅
29474 ⋅
3840⋅
Burnley
28228 ⋅
51262⋅
47
20⋅
Hull City
95535⋅
13269⋅
0817⋅
Arsenal
25496 ⋅
83415⋅
8837⋅
Crystal Palace
45337 ⋅
22302 ⋅
7919⋅
Everton
55456⋅
54380⋅
99
29⋅
Watford
05258⋅
60295⋅
3819⋅
Stoke City
28228 ⋅
71296 ⋅
53
21⋅
Leicester City
53
327⋅
68285⋅
51
21⋅
Manchester United
93406⋅
82447⋅
9632⋅
Southampton
05258⋅
70403⋅
5722⋅
Sunderland
10119⋅
13269⋅
5811⋅
Middlesbrough
43208 ⋅
07247 ⋅
29
14⋅
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231
The data on average distance (km) covered per game by each team and the
total distance covered by each team for the season was obtained from the Express
UK online (Express UK 2017). We found that the work rate index is substantially
smaller than the assist and key pass indices. Also, the goal difference index can be
either positive or negative. It can therefore add to or subtract from a team’s rating.
It is quite clear that the team contributions index carries the greatest weighting in
the overall rating index. Table 12 presents the overall rating calculated for the
twenty Premier League teams for the 2016/ 2017 season, arranged in order from
the highest to the lowest rating.
Table 12. Final Team Ratings
Position
Team
Team Rating
1. Chelsea 4807
2. Tottenham Hotspur 4616
3. Liverpool 4374
4. Manchester City 4173
5. Arsenal 4059
6. Everton 3445
7. West Bromwich Albion 3253
8. Bournemouth 3222
9. Manchester United 3194
10. West Ham United 3074
11. Swansea City 3039
12. Southampton 3019
13. Crystal Palace 3009
14. Stoke City 2822
15. Hull City 2788
16. Leicester City 2609
17. Burnley 2567
18. Watford 2542
19. Middlesbrough 2311
20. Sunderland 2076
Discussion
Recapping the optimal assignments for both teams, we observe that Chelsea’s
entire round total was higher that Manchester United’s. However, Manchester
United’s average defensive assignment was higher that Chelsea’s. As a result,
Manchester United conceded fewer goals than Chelsea, twenty-nine as opposed to
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thirty-three. Note that Chelsea scored significantly more goals than Manchester
United i.e., eighty-five to fifty-four. Chelsea also had a goal difference which was
more than twice that of Manchester United. Chelsea’s midfield and attack scored
approximately five and four more assignment points per game respectively. This
tells the story of how effective the link between midfield and attack worked for
Chelsea. The midfield also helped martial the defence by creating a formidable
barrier in front of the defence. We can see that the combined average assignment
per game in defence and midfield for Chelsea was 32.36 as opposed to 29.97 for
Manchester United. This shows how superior the midfield of Chelsea was in
supporting the defence and linking up the attacks.
This type of analysis can inform the manager and coaching as to the best
players for various roles. It will in fact aid in the selection of the team from the
available players - depending on the team formation adopted for a particular game.
For example, suppose that Chelsea was playing the (3-4-3) formation with three
defenders, having identified that the three most important roles to counteract the
opposition’s weaknesses were to intercept, clear and pass optimally to neutralize
the opposition’s attack. In such a case, Azpilicueta, Luiz and Terry would have
been the three best available options in that particular season.
It is interesting to observe that the top two and bottom two teams in our rating
index placed exactly the same as the final league table for 2016/2017. The top five
teams were also the same as the final league table, but with Manchester City and
Liverpool switching positions. We note that in our final team rating index,
Everton, West Bromwich Albion and Bournemouth all finished above Manchester
United, however this did not happen in the final league table for 2016/2017. This
was because offensively, their contributions on our ratings index were higher than
that of Manchester United.
Conclusion
The selection of individual football players to function as a cohesive unit can
be a very daunting task for coaches. Getting the right balance of strikers,
midfielders and defenders is critical to the team’s all-round performance. By using
data from previous games on how players perform the various roles, coaches can
explore the best combinations to use for upcoming matches. We have demonstrated
how this can be achieved via the application of the Hungarian Algorithm. Web
sources provide data on football statistics such as blocking, clearing, tackling,
intercepting, dribbling, shooting, assisting, passing, etc. We are able to divide
these attributes into defensive and offensive together with a combination of both to
pick the best defence, midfield and offence to perform optimally as a unit.
We have also described how to use player statistics to create a ranking system
for all registered teams in a football league. This can be achieved through the
creation of a team index by way of a combination of five sub-indices. The first
sub-index is called team contributions, and it accounts for the number of set piece
goals, shots, blocks, tackles won, aerial duels won, clearances, red and yellow
cards obtained by the players. For each team, the total number for each component
Athens Journal of Sports
December 2023
233
is multiplied by an estimated correlation coefficient and the resulting values are
added to determine the overall score that is representative of these contributions.
The four remaining sub-indices are called goal difference, assist, key pass and
work rate. Each of these sub-indices contributes a score to the overall team index,
based on the overall numbers that the team amasses in each respective aspect of
team play. The score for the five sub-indices is then totalled to produce the team
index score, and the teams are ranked from highest to lowest based on the final
index score.
We have used the 2016/2017 Premier League data to demonstrate the
similarities between our team ranking index and the eventual position of each team
in the league table at the end of the season. This suggests that our proposed team
index can be used as a league predictor for future seasons and to set up betting
odds for teams. Further analysis could be carried out to determine what proportion
each sub index contributes to the all-round team index. This would allow
conclusions to be drawn on the effectiveness of the various sub-indices and their
relative importance in predicting the outcome of the league.
The ratings index that we have presented in this paper provides an additional
tool for the comparison of teams. It allows us to analyse the overall performance,
and subsequently to determine the best and worst teams in the league. Some of the
ideas in creating this index were utilised and modified from (McHale et al. 2012).
The team index is a single score used to rate the collective player contributions
that directly influence overall team success. It provides a quantitative way to
measure the differences between teams.
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