Received May 24, 2021, accepted June 8, 2021, date of publication June 21, 2021, date of current version

June 30, 2021.

Digital Object Identifier 10.1109/ACCESS.2021.3091185

Multi-Objective Optimization for Football

Team Member Selection
HAOYU ZHAO1 , HAIHUI CHEN1 , SHENBAO YU 1, AND BILIAN CHEN 1,2
1 Departmentof Automation, Xiamen University, Xiamen 361005, China
2 Xiamen Key Laboratory of Big Data Intelligent Analysis and Decision-Making, Xiamen 361005, China

Corresponding authors: Shenbao Yu (yushenbao@outlook.com) and Bilian Chen (blchen@xmu.edu.cn)

This work was supported in part by the Youth Innovation Fund of Xiamen under Grant 3502Z20206049, and in part by the National
Natural Science Foundation of China under Grant 61836005.

ABSTRACT Team composition is one of the most important and challenging directions in the recommen-
dation problem. Compared with a single person, the advantage of a team is mainly reflected in the synergy
of team members’ complementary collaboration. To build a high-efficiency team, how to choose the team
members has become a tricky problem. However, there is a lack of quantitative algorithms and validation
methods for team member selection. In this paper, we put forward three indicators to measure a team’s
ability and formulate the selection of football team members as a multi-objective optimization problem.
Subsequently, an evolutionary player selection algorithm based on the genetic algorithm is proposed to solve
the team composition problem. We verify the effectiveness of the team member recommendation algorithm
via data analysis, football game simulation under different budget constraints and provide comparisons with
existing methods.

INDEX TERMS Team composition, multi-objective optimization, genetic algorithm.

I. INTRODUCTION football video games. In recent years, many football video

Teamwork is the collaborative effort of a team to accomplish games with high authenticity and reliability have been devel-
a common destination or to complete a shared task in the oped in the game market, such as Federation International
most effective and efficient way. Compared with a single Football Association (FIFA),1 Pro Evolution Soccer (PES),2
person, a team can integrate not just knowledge but skills of and Football Manager (FM).3 Such video games can assess
each member from different professional domains reasonably players’ attribute values precisely according to the real abil-
based on their characteristics. Teamwork can solve problems ities of those players on the pitch, which provides great
with complementary advantages and realize the value-added convenience for our research and verification, see the game
benefit of 1 + 1 > 2, which is critical for achieving common interface of PES2021 in Fig. 1. Moreover, take Electronic
goals. However, it is very difficult for us to form a team that Arts4 as an example, it is a famous interactive entertain-
covers all aspects of knowledge and skills needed, hence auto- ment software company in the world, which publishes the
matically recommending a team with high competitiveness, FIFA game series every year. Through the official website
by defining some indicators according to existing experiences https://sofifa.com/, we can obtain professional official ratings
and studying quantitative algorithms, is in great necessity. of more than 10,000 football players, such as players’ salaries
Football has become one of the most popular sports in the and nationalities. It has some advantages in the research of
world. At the same time, football is highly inherently cooper- team composition, such as high timeliness, full characteris-
ative, so football team composition is greatly representative tics, and rich content. Fig. 2 illustrates some official data of
of the studying of the team composition problem. In reality, the legendary football player Lionel Messi.
evaluating the effectiveness of football team composition The team member composition problem is quite differ-
is nearly impossible. However, fortunately, with the recent ent from the general single-objective optimization problem,
development and progress of electronic games, we can study
the football team composition problem by simulating virtual 1 https://www.fifa.com/
2 https://www.konami.com/
The associate editor coordinating the review of this manuscript and 3 https://www.footballmanager.com/
approving it for publication was Sotirios Goudos . 4 https://www.ea.com/

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

VOLUME 9, 2021 For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ 90475
 H. Zhao et al.: Multi-Objective Optimization for Football Team Member Selection

2021 (PES2021) platform, and comparison with other

classical approaches.
The remainder of this paper is organized as follows.
Section II introduces the latest progress of the recommenda-
tion system and its application in the field of sports team com-
position, as well as the literature review of multi-objective
optimization methods. We propose three indicators for evalu-
ating players and football teams in Section III. In Section IV,
we model the team member composition problem as a
multi-objective optimization problem and propose a modi-
FIGURE 1. Football game interface of the PES2021 game platform. fied genetic algorithm named ESP to solve it. We conduct
comprehensive experiments to validate the effectiveness of
our proposed ESP algorithm through data analysis and game
simulation in Section V. Section VI compares our proposed
model and the corresponding algorithm with others. Finally,
we conclude our work, discuss the shortages and give some
future directions in Section VII.

II. RELATED WORK

A. TEAM RECOMMENDATION IN SPORTS FIELD
The research on recommendation systems can be traced back
to the mid-1990s. Its main application is to recommend items
or services to users based on their similar preferences, known
FIGURE 2. Player profile of Lionel Messi. as collaborative filtering. Recommendation system plays a
pivotal role in online shopping, e-commerce services, and
social network applications. In recent years, there has been
which has only one specific objective to optimize. However, a wide range of practical issues covering researches and
the team composition is often affected by multiple aspects, developments in the field of recommendation. For exam-
which makes the evaluation become very subjective and vary ple, Ayata et al. [1] proposed an emotion-based music rec-
with personal preferences, hence the simple optimization ommendation framework that learns user emotions from
methods and recommendation algorithms failed in this sce- wearable physiological sensor signals. Sun and Zhang [2]
nario. But fortunately, we can measure the ability of a team integrated techniques in dialog systems and recommender
through various evaluation indicators, which are then turning systems into a novel and unified deep reinforcement learning
into different objectives by considering different aspects of framework. Furthermore, Strub et al. [3] enhanced the hybrid
team members. By optimizing different objective functions, recommender systems based on Autoencoders. However, few
we simplify the practical team member composition problem studies have applied related conceptions and technologies
into a multi-objective optimization problem, which can be of recommendation systems to the problem of football team
solved by established techniques. member composition, let alone the automatic team compo-
In this paper, we conduct a specific and comprehensive sition applied in the field of electronic virtual football video
study on the topic of the team member recommendation games.
problem. We first define some conceptions that are useful On recommendation for sports, Qader et al. [4] pro-
for elaborating our novel indicators, and then adopt modified posed a method for evaluating and ranking football players
Non-dominated Sorting Genetic Algorithms-II (NSGA-II) to based on multi-criteria decision-making, where players were
compose a high-quality football team in terms of winning selected according to several physical fitness indicators, such
rate. Finally, we implement the proposed method and evaluate as 30-meter speedrunning. Similarly, Di Salvo et al. [5]
it through the simulation of the football video game PES2021. investigated performance characteristics according to skill
We list the main contributions as follows: positions of elite soccer players. In the study, they argued
• We define three indicators to evaluate the performance that different positions have different physical requirements
of football players, which contribute to a novel frame- for players. In [6], Özceylan adopted the Analytic Hierarchy
work for team composition. Process (AHP) algorithm, which is a structured technique for
• We formulate the team composition problem as a novel group decision making, combined with 0-1 integer program-
multi-objective optimization function, and propose a ming method, to select players. Kamble et al. [7] also applied
modified genetic algorithm named Evolutionary Selec- the AHP algorithm to select cricket players. Despite the pop-
tion of Players (ESP) to solve it. ularity of the AHP, many authors have explored some other
• We evaluate the proposed model via numerical analysis, heuristic methods for player selection. In [8], Ahmed et al.
game simulation based on the Pro Evolution Soccer utilized the NSGA-II algorithm to select players, but the

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definition of the objective function is subjective, and there and particle swarm optimization [16], [17]. Particularly,
is no reliable verification method to prove the effectiveness MOEA/D, an acronym for multi-objective evolutionary
of the algorithm. Grund [9] exploited the density of network algorithm based on decomposition, provides a new idea
data structure, taking the Premier League football team as for solving MOPs [18]. Inspired by MOEA/D, some
an example, to study the team performance. However, if the decomposition-based algorithms are also proposed [19]–[21].
network structure is used to connect players in the same team Furthermore, evolutionary algorithms also provide new ways
for player recommendation, there will be a problem that all for solving practical problems. For example, Li and Yin [22]
players cannot be connected by each other, i.e., the player used differential evolutionary algorithm to design a recon-
graph is not connected. Besides, a fuzzy inference system figurable antenna array with quantized phase excitations.
is also adopted into player selection [10], but this method In [23], a multi-search differential evolutionary algorithm
largely depends on the experiences and cognitive abilities with self-adaptive parameter control was proposed for solv-
of experts. Zeng et al. [11] hashed over a skill coverage ing global real-parameter optimization problems. Addition-
function by using the submodule optimization technique, ally, several heuristic optimization algorithms have also been
which selects players by maximizing the constructed skill explored to deal with the MOPs. For instance, Li et al. [24]
coverage function. Nevertheless, the constructed function still proposed a new heuristic optimization method, called animal
has some shortcomings, e.g., the skill coverage of a team will migration optimization algorithm. The algorithm is inspired
reach a maximum as long as one player has a very high score by animal migration behavior, which is a ubiquitous phe-
on a certain attribute, no matter how other players perform. nomenon that can be found in all major animal groups.
Besides, it is not suitable to predict players’ salaries based Besides, Li et al. [25] put forward a new multi-objective
on the fitted exponential function which only considers the forest algorithm that can identify protein-RNA interactions
performance-price ratio of players. from CLIP-seq data. This study provides a refreshing insight
Team composition is often more than a temporary consid- into the use of multi-objective optimization for genome
eration. For example, on the football field, player transactions informatics.
and contracts would not expire within four years. Therefore, In addition to the MOPs, many of the previous
the potentials of team members are of great importance. researches on dynamic multi-objective optimization prob-
Teams with greater potentials tend to achieve higher valua- lems (DMOPs) [26]–[28], which focuses on the multiple
tions and better long-term performance. There are a variety conflicting goals that change over time, have also been
of literature on the field of evaluating potentials of football explored. For example, Xu et al. [27] proposed a cooperative
players. E.g., Williams and Reilly [12] attempted to integrate co-evolutionary strategy based on environmental sensitivities
their main research findings with talent identification and for solving DMOPs. Furthermore, Rong et al. [26] put for-
development in soccer. Similarly, Unnithan et al. [13] did ward a multi-directional prediction strategy to enhance the
research on talent identification in youth soccer. In [14], performance of EAs for DMOPs.
Jimǹez and Pain explored the relative age effect in Spanish In this paper, we focus on the genetic algorithm, one of
Association football and provided a comprehensive elabora- the pivotal methods in EAs, which aims to balance different
tion on the influence of player ages. objective functions as well as find the solution set that makes
However, in the studies mentioned above, the indicators each objective function as optimal as possible. Among several
of selecting players are subjective and lacking quantitative multi-objective genetic algorithms, NSGA [29] is one of the
evaluation. The subjective player selection methods cannot most influential and widely used ones, and has been improved
be verified by numerical approaches. Besides, those methods by Deb et al. [30]. The improved one is called NSGA-II. Due
only focus on football players’ physical fitness, ignoring to the simplicity and effectiveness, the NSGA-II algorithm
the comprehensiveness of the composed group. Furthermore, has successfully been applied in various fields [31]–[33].
those researchers did not consider the potential of the com- Based on the relationship between football players and
posed team, i.e., the future performance of the team, which is teams, we propose three indicators to evaluate the ability of
of great importance. a team. Besides, we construct the team composition problem
as a novel multi-objective optimization function, which will
B. MULTI-OBJECTIVE OPTIMIZATION METHODS be resolved by a new modification of NSGA-II algorithm and
Multi-objective optimization methods have been widely we further verify the effectiveness of our proposed model and
used in many areas, including engineering and economics algorithm by simulating virtual video game.
where optimal decisions need to be taken in the pres-
ence of trade-offs between two or more conflicting objec- III. ESTABLISHMENT OF EVALUATION INDICATORS
tives. In the past decades, numerous studies have attempted In this section, we design three indicators to evaluate the per-
to solve multi-objective optimization problems (MOPs). formance of teams and football players, which contribute to a
Among them, evolutionary algorithm (EA) is one of the novel framework for team composition. The reason why we
mainstream algorithms. Many researches have been carried consider the three indicators will be discussed in Section V
out, including ant colony algorithm [15], genetic algorithm in detail.

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A. OVERALL EVALUATION OF PLAYERS AND TEAMS as many as 26 positions on the pitch, to save space, readers
Given a football player Pi , the most important factor for team may refer to the website https://sofifa.com/ for more specific
composition is the player’s ability, which we define as the abbreviations.
overall evaluation φOverall (Pi ). It is a comprehensive property
based on the player’s general performance in a football game. C. PLAYER POTENTIAL
The overall evaluation of a player considers different football The two factors mentioned above are based on the current
skills, including the players’ physical attributes, football tech- situation, but if we want the result of team composition to
nology, and psychological quality. We integrate the overall be effective, workable, and sound, it must be grounded in
evaluations of all football players in a team to form the team the present and the future as well. For a football player,
ability, which is defined as φOverall (N ), where N is the total we describe future ability as the potential attribute. However,
number of football players in a team. The specific calculation how to measure the potential value of players in a team is a
method can be seen in Eq. (1). difficult problem. According to the analysis of the players’
N
data, we found that the characteristics associated with the
X potential value of players are players’ age, current perfor-
φOverall (N ) = φOverall (Pi ) (1)
mance, and so on. Given a player Pi , we use Po(Pi ) to rep-
i=1
resent his potential value, which consults the comprehensive
B. OFFENSIVE AND DEFENSIVE ABILITY OF PLAYERS evaluation results of football scouters in the dataset.
It is not enough to form a competitive football team only
based on a player’s overall evaluation, which will evoke some IV. TEAM COMPOSITION AS A MULTI-OBJECTIVE
shortcomings. For example, if the selected players are all for- OPTIMIZATION PROBLEM
wards, it will make the team’s defensive ability insufficient. As previously asserted, building a football team with a high
As we have analyzed above, there is a significant difference winning rate requires many considerations, including the
in soccer players’ abilities at different positions on the field. overall evaluation of the team, the offensive, and defensive
When measuring the ability of a forward, we require more abilities of the players, as well as the potential value. In this
offensive skills, such as control and speed of the ball and section, we model the team composition as a multi-objective
shooting skills. Similarly, when considering the ability of a optimization problem based on the three factors proposed in
guard, we desire higher defensive attributes such as physical Section III. We first formulate the football team composition
contact ability. Taking the famous football star Messi as an problem, and then elaborate on the algorithm for selecting
example, his ability in an offensive position is generally players.
higher than that in a defensive position, so it will be sensible
to put this player in the offensive position. A. MODEL FORMULATION
In this paper, we consider the football players’ abili- We formulate the team composition problem as a
ties for different positions, and divide the positions except multi-objective optimization problem based on a series of
for Goalkeeper into three parts: Attack position (e.g. Striker, optimization parameters. Given a football player Pi , let S(Pi )
Center Forward), Midfield position (e.g. Center Midfield) represent the player’s salary, and B be the team’s total budget,
and Defensive position (e.g. Center Back). Attack posi- the multi-objective optimization problem can be formulated
tion describes a player’s offensive ability, while Defensive as follows:
position measures the player’s defensive attribute. Besides,
X
N
 φOverall (Pi )
we consider both offensive and defensive abilities for

Xi=1


 N
the Midfield position because of its position specialty.


 Po(Pi )
For a soccer player Pi , we use λAtk (Pi ) and λDef (Pi ) to Xi=1


Nplayers
represent a player’s offensive ability for Attack position and max λAtk (Pj )
j=1
defensive score for Defensive position respectively, and λGK

 X Nplayers
λDef (Pj )



refers to a player’s goalkeeping attribute. The calculation

 j=1
N

GK
X
λGK (Pk )

method of a player’s offensive ability is the average score


k=1
of his abilities in different offensive positions (and similar N
applies to the measure of defensive ability). We adopt the
X
s.t. S(Pi ) 6 B (3)
following method to compute them: i=1
(
λAtk (Pi ) = Mean(µSt , · · · , µCam ) where Nplayers is the total number of football players in three
(2) positions (i.e. Attack position, Midfield position, and Defen-
λDef (Pi ) = Mean(µCb , · · · , µCdm )
sive position) and NGK is the number of goalkeepers. The
where µ represents the player’s performance in different parameters N , Nplayers and NGK are all constants. Particularly
positions. For example, µSt shows the player’s performance in our case, we have Nplayers = 10, NGK = 1, and N =
in the Striker (St). Similarly, µCam is the player’s perfor- Nplayers + NGK = 11. Indeed, the proposed model formulated
mance in the Center attack midfield (Cam). Since there are in Eq. (3) can be extend to a matchday team with reserves

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or an entire team. For example, when considering the bench

players, we can adjust the total number of football players N
as needed.
Note that we formulate the team composition problem
as a multi-objective optimization shown in Eq. (3). In the
single-objective optimization problem, we can easily deter-
mine the superiority of a solution over other solutions by
comparing their objective function values. However, for the
multi-objective optimization problem, the result is a set of
solutions that achieves the best trade-off between competing
objectives, and the goodness of a solution can be obtained by
the Pareto dominance [34].

B. ESP ALGORITHM
There exists multiple Pareto optimal solutions for multi-
objective optimization, and evolutionary algorithms funda-
mentally operate on a set of candidate solutions, thus we focus
on genetic algorithms, which is one of the major evolutionary
algorithm paradigms, for further solving the optimization
problem. In this section, we elaborate on the ESP algorithm,
which is a modification of NSGA-II. We first provide a brief
description of the NSGA-II procedure [30] and then illustrate
the ESP algorithm in detail.
Fig. 3 shows the general procedure of NSGA-II. The
basic flow of NSGA-II algorithm is similar to the tra-
ditional genetic algorithm, including critical components
such as coding, crossover, mutation, and selection. Besides,
NSGA-II explores three special characteristics (i.e., fast
non-dominated sorting approach, density estimation and
crowded-comparison operator). Specifically, according to the FIGURE 3. The flowchart of the NSGA-II algorithm.
criteria for the sorting process, NSGA-II first initializes a ran-
dom parent population. The population is sorted based on the TABLE 1. A toy example of the chromosome reordering process.
non-domination. Once the first sorting is completed, the usual
binary tournament selection, recombination, and mutation
operators are used to create an offspring population, which
is then combined with the current generation population.
Followed by the combination procedure, NSGA-II introduces
In addition, we also arrange the remaining 10 bits of
the elitism criterion to compare the current population with
the chromosome in ascending order according to the serial
the previous best solutions and selects the individuals of the
number, which represents the player’s salary (see Table 1).
next generation based on the crowded-comparison operator.
The smaller the value of the serial number, the higher the
Based on the multi-objective player selection optimization
player’s salary. During the cross recombination procedure,
model, the ESP algorithm absorbs the advantages of NSGA-II
rearranging the similar chromosome sequence makes a small
and some modification is done to make it better fit the need
difference in salary between the two players, and avoids
for the football player selection problem. Specific changes
meaningless updating operations. Furthermore, it can better
are explained from the following two aspects.
control the total cost of the team and reduce the probability
1) CODING METHOD
of non-feasible solutions.
In the genetic algorithm NSGA-II, the individual variables of
2) FAST NON-DOMINATED SORTING PROCESS BASED ON
each generation are usually continuous. However, we make
BUDGET CONSTRAINTS
each variable of individual as an integer representing the
serial number of each player. Besides, the length of a chro- It is not an easy task to solve the team composition problem
mosome is set to 11, which equals the number of football with budget constraints. In the ESP algorithm, we introduce
players. Considering there is only one goalkeeper in a team, a variable Constraint Violation (CV) in Eq. (4) to make the
we choose the first bit of the chromosome limited to the goal- final solution satisfy the budget constraint.
 
keeper selection and the remaining 10 bits of a chromosome TC − B
CV = max 0, (4)
represent non-goalkeeper players. B
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where TC is the total cost of the team. Based on the above TABLE 2. An example of football players’ athletic abilities.
formula, the constraint violation value of a feasible solution is
always 0, while the one for the non-feasible solution is greater
than 0. The larger the violation value, the greater the deviation
of the non-feasible solution.
By introducing the concept of the constraint violation,
we change the rules of the fast non-dominated sorting pro-
cess, which is the key component of the NSGA-II algorithm.
Given any two solutions (i.e. individuals in the population) xa TABLE 3. An example of football players’ personal attributes.
and xb , of which decision values are a , CV (xa ) and
b , CV (xb ) respectively, the new dominant relationship is
obtained as follows:
• If CV (xa ) = 0 and CV (xb ) > 0, we have a dominates b,
or otherwise, b dominates a.
• If CV (xa ) > 0 and CV (xb ) > 0, we have the smaller CV
dominates the larger CV . We first perform data preprocessing by filtering out some
• If CV (xa ) = 0 and CV (xb ) = 0, the dominant relation- irrelevant attributes and sort the data (in descending order)
ship is determined according to the rules mentioned in according to players’ salaries. Table 2 and Table 3 list some
Pareto dominance. players’ basic properties after data preprocessing. We then
Algorithm 1 presents the details of the player selection pro- elaborate on the three phenomena discovered from the exper-
cedure for the ESP algorithm. We initialize a set of candidate imental data, which consist of our assumptions.
players P, a budget constraint B, and the hyperparameters
such as mutation probability pm , crossover probability pc , Algorithm 1 Finding a Best Team Based on ESP Algorithm
and polynomial mutation distribution index ηm . After sorting Input: crossover probability(pc ), mutation probability(pm ),
players based on their salaries, the initial population of a given polynomial mutaion parameter(ηm ), player dataset(P),
size is randomly generated by integer coding (Line 1 - Line 7). player number(N ), budget(B).
The first generation of offspring population is obtained by Output: optimal solution set(P̂).
basic operations of crossover and mutation of genetic algo- 1: pop = ∅.
rithm (Line 8 - Line 11). Starting from the second genera- 2: // Individual coding and population initialization
tion, the parent population and the offspring population are 3: for i = 1 to PopSize do
merged, followed by calculating the CV values and perform- 4: RandomSequence = randomly choose the index of N
ing the fast non-dominated sorting process with constraints. players from P.
At the same time, we compute the crowding degree of the 5: individual = RealNumberCoding(RandomSequence,
individuals in each Pareto front. Note that among the two budget).
solutions with different Pareto fronts, we prefer the solution 6: pop.add(individual).
with a better dominance ranking. Otherwise, if the two solu- 7: end for
tions belong to the same front, we prefer the solution in the 8: for i = 1 to Max_Generation do
relatively less crowded region. According to the crowding 9: newpop = CrossOver(pop, pc ).
degree of individuals, the appropriate individuals are selected 10: newpop = Mutation(newpop, pm , ηm ).
to form a new parent population (Line 12 - Line 17). The algo- 11: newpop = newpop + pop.
rithm loops until the conditions for the end of the program 12: // Calculate constraint violation for each individ-
are met. ual
13: CV = ConstraintViolation(newpop, budget).
V. DATA ANALYSIS AND EXPERIMENTS
14: newpop = FastNondominantSort(newpop, CV ).
We implement the algorithms in Python 3.85 and conduct
15: crowding = CrowdingCompare(newpop).
all the numerical computations on a Windows PC with a
16: offspring = Selection(newpop, crowding).
4-core Intel i5-1135g7 2.40GHz CPU and 16GB memory.
17: pop = offspring.
All the experimental data is collected from the website
18: end for
https://sofifa.com/ and all the games are simulated in a quick
19: return P̂.
game of PES2021.
A. DATA ANALYSIS
1) WAGES ARE NOT DIRECTLY PROPORTIONAL TO
In this section, we analyze the experimental data and present
PLAYERS’ ABILITIES
some facts related to our proposed multi-objective function.
We first analyze the relationship between the player’s ability
5 Part of the code and dataset are available from: https://github.com/haoyu- and wages. Taking the player’s overall rating as the X-axis,
zhao/Multi-Objective-Optimization-for-Football-Team-Member-Selection the corresponding weekly salary as the Y-axis, we plot the

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H. Zhao et al.: Multi-Objective Optimization for Football Team Member Selection

FIGURE 4. Players’ wage in the different overall ratings.

relationship in Fig. 4, each dot represents a football player.

We can observe that under the same overall ratings, football
players have different salaries. This finding is consistent with
that we cannot simply measure a player’s contribution to a
team only based on his salary.

2) POTENTIAL PLAYS AN IMPORTANT ROLE IN FUTURE

PERFORMANCE
As described in Section III-C, we consider players’ potential
value as it plays an important role in team composition. Fig. 5
shows the changing trend of the average value for poten-
FIGURE 5. The relationship between age and potential value of players in
tial capability and overall evaluation of all players with the different leagues.
increase of age in different football leagues. We observe that
the overall evaluation and potential value of young players
are quite different initially, while with the growth of age,
the potential of future players will be basically consistent with
the overall evaluation of players. It means that the potential
value of young players can reflect their future performance.
The higher potential value may result in a higher overall
evaluation in the future. Therefore, it is necessary to consider
the player’s potential value for football player selection.

3) THERE IS NO PERFECT PLAYER

From the processed data, we observe that there is no per-
fect football player. We choose three representative attributes
from 20 iconic players, including players’ overall evaluation,
potential value and salary as shown in Fig. 6. Results show FIGURE 6. The overall evaluation, potential value and salary of 20 iconic
that no player is better than other players in all three attributes. players.
We conclude that it is impractical to simply select the player
who reaches the best performance in all aspects. It needs a of the ESP algorithm are set as follows: pc = 0.5, pm =
trade-off among the three indicators in Section III, including 0.1, ηm = 30, PopSize = 400, and Max_Generation =
the player’s overall evaluation, the offensive and defensive 300. The mutation mode of the chromosome is polynomial
ability, and the potential value. mutation [35].
We first visualize the results of the ESP algorithm under
B. EXPERIMENT RESULTS different budget constraints. Fig. 7 shows all individuals in the
To verify the strength of the team generated by the proposed last generation population without budget constraints. Simi-
method, we use our ESP algorithm to solve the optimization larly, we depict all solutions under a certain budget in Fig. 8.
problem under two different situations: team composition The horizontal axis and vertical axis represent the team’s
with or without budget constraints. Furthermore, we also use average offensive ability and defensive ability respectively,
the t-test to demonstrate the effectiveness of our algorithms. and the Pareto optimal solutions are marked in red dots.
We manually set the bounds of parameters, and then adopt We can observe from Fig. 7 and Fig. 8 that the Pareto optimal
grid search technique to find the optimal parameters based on solution set achieves better attacking ratings and defensive
the simulation time and performance. Hence, the parameters ratings than other solutions.

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FIGURE 7. Pareto solutions for ESP algorithm without budget constraints. FIGURE 9. The influence of budgets on Pareto solutions.

TABLE 4. Comparison of players’ average athletic abilities between the

ESP Dream Team and Random Team without budget constraints.

TABLE 5. P-Value (without budget constraints).

FIGURE 8. Pareto solutions for ESP algorithm with budget constraints.

Besides, we analyze the Pareto fronts under different bud- by the ESP algorithm has certain numerical advantages over
get constraints, as shown in Fig. 9. The green dot set is other teams without budget constraints.
the Pareto solutions without budget constraints. In Fig. 9, In the t-test settings, given a team attribute (i.e. overall
we see that with the increase of the budget, the Pareto front evaluation, attacking rating, defence rating, or goalkeeping),
shifts to the lower-left corner, that is, the higher the budget, we assume that the team generated from the ESP algorithm
the stronger the abilities of attack and defense of the team, and the random team share the same average value. We first
which is consistent with our assumptions. take out 200 teams selected by the ESP algorithm, as well
As can be seen in Fig. 9, there are many candidate solu- as 200 random teams to simulate the distribution of team
tions, which form a Pareto front under the same budget. attribute values with large samples. For each type of teams,
By descending sorting the crowding degree of the solutions we then draw a histogram by calculating the frequency of
in Pareto fronts, we obtain the optimal solution (i.e., the best attribute scores in Fig. 10, and the corresponding P-values
team). Unless otherwise indicated, we use ESP Dream Team are shown in Table 5. Based on the P-value of all attributes,
to represent the best team in our simulation experiments. we can reject the original hypothesis with confidence that the
average values of any team attribute of both the ESP team
1) BUDGET UNCONSTRAINED CASE and random team are the same, which in turn demonstrates
In this section, we analyze the selection results of the ESP the effectiveness of our algorithm.
algorithm without budget constraints. We set a large num-
ber, which equals 770 million euros to simulate a sufficient 2) BUDGET CONSTRAINED CASE
budget. Table 4 provides a comparison of players’ average We compare the numerical performance of our team with a
athletic abilities between the ESP Dream Team and a random randomly selected team under the constraints of the budget
team under a similar total budget level. The players’ average in this section. We set the budget at 150 million euros, which
athletic abilities include the average of the overall evaluation, is a representative budget of a football team. The comparison
attack rating and defense rating, as well as the goalkeeper’s results are shown in Table 6. It can be seen that all the
goalkeeping ability. In Table 4, we see that the team selected numerical results of our proposed method are better than

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TABLE 7. P-Value (with budget constraints).

TABLE 8. Comparison of players’ average athletic abilities between the

ESP Dream Team and Random Team with budget constraints considering
player potential value.

FIGURE 10. Histogram of the distribution of four attributes without

budget constraints.

TABLE 6. Comparison of players’ average athletic abilities between the

ESP Dream Team and Random Team with budget constraints.
TABLE 9. P-Value (with potential considerations).

FIGURE 12. Results of players’ average athletic abilities of the ESP Dream
Team and Random Team.
FIGURE 11. Histogram of the distribution of four attributes with budget
constraints.
football team. Table 8 shows the optimal solution selected
that of the random values. From Table 6, it is clear that the by the ESP algorithm when the budget is 100 million euros.
ESP algorithm delivers the best performance in all aspects It can be seen that with a similar budget, the ESP algorithm
while using fewer salaries. Similar to the t-test settings in considering the players’ potentials can select a team with
Section V-B1, we show the distributions of all attributes’ excellent potential value while keeping the four better men-
average values and P-values based on the budget constrained tioned indicators. To better understand the proposed method,
case in Fig. 11 and Table 7 respectively, and the correspond- Fig. 12 gives a preview of two team’s properties, where each
ing results also demonstrate the effectiveness of our proposed dimension shows a kind of average ability. From Fig. 12,
algorithm. we can also see that the team selected by the ESP algorithm
is better than the team generated from the random algorithm
3) POTENTIAL IMPACT in all aspects. Likewise, we also show the distributions of all
It is hard to verify the influence of the potential attribute attributes’ average values and P-values when considering the
for the football player selection in reality because we can potential effect. The corresponding results (see Fig. 13 and
only observe the current performance of players. Therefore, Table 9) demonstrate the effectiveness and rationality of the
we provide the numerical result of the potential value for a proposed factors in our modeling.

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TABLE 10. Game simulation results of the ESP Dream Team v.s. Random Teams.

TABLE 11. Game simulation results of the ESP Dream Team v.s. Ten
Representative Real Teams from different football leagues.

FIGURE 13. Histogram of the distribution of four attributes without

budget constraints considering potential considerations. To compare the algorithm performance, we randomly select
teams without any constraint and simulate the battle between
the two teams. The match results are listed in Table 10,
C. GAME SIMULATION RESULTS
including specific results of every match, the total score and
The numerical results cannot reflect the match level of a the Goal Difference. As can be seen from Table 10, we win 29
football team. In this section, we show the competition results out of 30 games, which indicates the ESP Dream Team is
of football simulation games using the PES2021 platform to dominant in most of the matches, which is exactly what we
verify the actual effectiveness of our algorithm. Two metrics expected.
are used for evaluating the results, one is Goal Difference, Besides, we make our ESP Dream Team against ten teams
which refers to the net wins, the other is Total Score defined that have the leading record in their respective football
as follows: leagues, such as Real Madrid CF6 and Manchester City FC.7
let g = {g1 , g2 , · · · , gM } be a competition result set, M is Results are shown in Table 11, we can see that in the face
the number of matches, we use Tps to represent the total score of different teams, the ESP Dream Team still achieves bet-
of the team in Eq. (5): ter performance. Furthermore, we choose two representative
M
X teams (i.e. FC Barcelona8 and Paris Saint German F.C.9 )
Tps = ωgi (5) and then battle with the ESP Dream Team, respectively. The
i=1 corresponding results are displayed in Table 12. Similarly,
where ωgi is the score of a match gi and its definition is in it can be seen that the ESP Dream Team wins most games
Eq. (6): with vastly superior forces, which dominates most of the
 competitions.
3
 Win
ωgi = 0 Draw (6) 2) BUDGET CONSTRAINED CASE

−1 Lose

Similar to the unconstrained numerical experiments, we test
Followed by the procedure in Section V-B, we verify the the effectiveness of our algorithm under a constrained case.
performance of the proposed method under different budget We compose the ESP Dream Team under the budget of 150
constraints. million euros and conduct thirty soccer matches with two
6 https://www.realmadrid.com/
1) BUDGET UNCONSTRAINED CASE 7 https://www.mancity.com/
We first compose the ESP Dream Team based on 8 https://www.fcbarcelona.com/
unconstrained budgets for the PES2021 game simulation. 9 https://www.psg.fr/

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H. Zhao et al.: Multi-Objective Optimization for Football Team Member Selection

TABLE 12. Game simulation results of the ESP Dream Team v.s. Real Teams.

TABLE 13. Game simulation results of the ESP Dream Team (with budget constraints) v.s. Random Teams.

TABLE 14. Budget level. Dream team, which indicates the stability and reliability of
the proposed algorithm.

VI. METHOD COMPARISON

In this section, we compare our model with other approaches
from two perspectives. One is the different ways of team
composition methods, the other is the different methods for
solving the multi-objective optimization problem.

A. COMPARISON OF TEAM COMPOSITION METHODS

random teams with similar budget constraints respectively, In this part, we present the comparison results between our
and the corresponding match results are shown in Table 13. team composition strategy and the one proposed in [11].
Despite lower goal differences compared to that in Table 10, In [11], the authors converted the team composition into a
the ESP Dream Team still outperforms the two random teams submodular optimization problem and proposed an algorithm
with a similar total budget. called CEFG (Cost-Effective Forward selection Greedy) to
Following the simulation competition with random teams, solve it. We again use the PES game platform mentioned in
we also fight the ESP Dream Team with real teams under dif- Section V to compare our ESP with the CEFG algorithm.
ferent budget levels (see Table 14). We first select several bud- We first record the best team generated by the ESP and CEFG
get levels, for each budget level, we generate the ESP Dream respectively, then simulate 30 game battles between these
Team by the proposed method and pick up a representative two selected teams. The simulation results are summarized
real team under the similar budget level. We then simulate 30 in Table 16. It can be seen from this table that among all 30
battles between two teams as shown in Table 15. Notice that matches, the team selected by our ESP algorithm loses only
the simulation results cover different budget levels, includ- three games and achieves good performance in goal differ-
ing teams with lower total salaries, such as Everton10 and ence, which demonstrates the effectiveness of our method.
Olympique Lyonnais,11 as well as teams with moderate and A possible explanation for this observation might be that
higher budgets, such as Juventus12 and PSG. Specifically, the team composition approach proposed in [11] ignores the
there are two observations from the table. First, all four ESP players’ positions and the future ability of a team, while both
Dream Teams win the most matches against real teams under of which are incorporated in our method.
the same or even higher budget level, which demonstrates that
the ESP algorithm generates promising performance. Second, B. COMPARISON OF MULTI-OBJECTIVE OPTIMIZATION
no matter how much we limit our budget, there is almost no METHODS
change to the numerical value of Tps achieved by the ESP In addition to NSGA-II, there are several algorithms on
solving multi-objective optimization problems. Here we
10 https://www.evertonfc.com/ select another classical multi-objective optimization evolu-
11 https://www.olweb.fr/ tionary algorithm (i.e., MOEA/D) [18] for comparison.
12 https://www.juventus.com/ MOEA/D transforms a multi-objective optimization problem

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TABLE 15. Game simulation results of the ESP Dream Team (with budget constraints) v.s. Real Teams.

TABLE 16. Game simulation results of the ESP Dream Team v.s. CEFG Dream Team.

VII. CONCLUSION AND FUTURE WORK

In this paper, we give three evaluation indicators for football
team composition, including overall evaluation, offensive and
defensive abilities, and players’ potential value. We formulate
the team composition issue as a multi-objective optimization
problem and propose a variant of the genetic algorithm named
ESP, which can automatically output and recommend a foot-
ball team with a high winning rate by quantifying the players’
abilities under a certain budget constraint. We also discuss the
FIGURE 14. Performance comparison between the ESP and modified effectiveness of our approach and the results demonstrate the
MOEA/D algorithm.
strength of teams generated via the proposed approach.
Despite these satisfying results, there is still room for
into a number of scalar quantum sub-problems and each improvement. For example, when recommending football
sub-problem is composed of a uniformly distributed weight team members, the existing ESP algorithm has no subjective
vector, which helps to generate the single objective function. and emotional input, as well as the lack of consideration of the
In order to apply it for our team composition problem, decision makers’ personal preferences. For instance, a foot-
we modify the MOEA/D algorithm, which mainly includes: ball team coach who values the team’s defense may choose
• Changing the coding method: we set the solution as a
players with higher defensive ability. To solve this problem,
vector of players whose length is equal to the number of we can develop a customized GUI (Graphical User Inter-
team members; face) based user-friendly software for users to personalize all
• Adding constraints: for each iteration, the solution gen-
attributes of the team composition problem. Besides, another
erated from MOEA/D method needs to be repaired until work is that the program developed here can be extended to
it satisfies the budget constraint; other similar sports fields, which we will postpone as a recent
• Target normalization: when using the MOEA/D, it is
study.
necessary to normalize the values of attributes men-
tioned in Section III.
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