The Role of Chance in Individual Sports: an Agent-Based Approach for Fencing Tournaments

It is a widespread belief that success is mainly due to innate qualities, rather than to external forces. This is particularly true in sport competitions, where individual talent is considered the only ingredient in order to reach success. In this study, we propose to explore the relative weight of talent and luck in individual sports through agent-based models. In particular, we chose fencing as case study, that is a combat sport involving a weapon. Fencing competitions are structured as direct elimination tournaments, where randomness is explicitly present in some rules. Our dataset covers the last decade of international events and consists of both single competition results and annual rankings for male and female fencers under 20 years old (Junior category). We show that our agent-based approach, calibrated on the dataset and parametrized by just one free variable a describing the importance of talent in competitions ( a = 1 indicates the ideal scenario where only talent matters, a = 0 the complete random one) is able to reproduce the main stylized facts observed in real data, both at the level of single fencing tournaments and of entire careers of a given community of fencers. We ﬁnd that simulations approximate very well the real data when talent weights slightly more than luck, i.e. when a is around 0 . 55 for Junior Men, or even slightly less than luck, i.e. when a = 0 . 45 for Junior Women. We conclude that the role of chance in fencing is highly underestimated even if probably it represents an extreme case for individual sports. Our results shed light on the importance of external factors in both athletes’ results in single tournaments and in their entire career, making even more unfair the “winner-takes-all” disparities in remuneration.


Introduction
Thinking about successful careers in general, and in sports in particular, the common belief is that they are just the result only of hard work, endurance and effort. One could naively think they develop from a long series of successes in several competitions, after endless training sessions and great sacrifices, and that only predestined champions with innate uncommon abilities can get them. Inner talent, in this view, makes the main difference in career development. However, many recent studies have shown that talent does not vary so widely among people and that its probability distribution is limited and concentrated around a well defined mean value. [1][2][3][4] Although individual talents do not differ that much from one person to another one, only a few reach the top. We experience it in many fields [4][5][6][7] and clearly in sports 3,8 , where we observe how people suddenly get notoriety and often huge amounts of money when they win.
Even with comparable talents, usually very high, athletes could end up with totally different rewards in a competition. One event after another, small fortuitous differences might give rise to a cumulative advantage, 9,10 which generate a consistent increasing gap between individuals with similar talents. We tend to admire those who reach the top, disregarding others in the ranking. We are so used to this common "winner-takes-all" logic that it sounds inevitable, in sports, arts, even in science. And most often the rationale given for such selectivity is formulated in terms of innate talent, coupled with effort: "they are the best".
We still lack a clear understanding of the processes which lead to vastly different rewards in sports and many other settings. In reality many small and unpredictable circumstances often play a role, but very often we tend to ignore these influences, preferring to believe in truly exceptional or gifted people. This talent oriented view persists despite being rejected by a wealth of evidence. 2-7, 11, 12 In this paper, we investigate these effects in the context of individual sports, where it is easier to analyse results and then assess the consequences. Moreover, some of them have simple rules and act in a controlled environment, suitable for testing an agent-based model which can reproduce athlete performances, capturing the role of individual abilities versus external circumstances in achieving success.
We will specifically consider fencing, as it is made up of face-to-face matches (called bouts) that directly compare athletes, underlining their similarities in contrast with huge differences in their outcome. Moreover, it contains rules where randomness is involved (for example, in case of tie). Fencing is a combat sport involving a weapon, which can be of three different kinds, identifying three separate disciplines: épée, foil and sabre. We focus on épée because there is no right of way rule regarding attacks, which means that any hit is counted. 13,14 Fencing is a perfect example of how unpredictable factors can strongly condition careers. First of all, randomness is explicitly present in its rules: in case of a tie, one extra minute is given, assigning a priority at random to one fencer. Secondly, competitions are arranged as tournaments with two distinct phases (pools and direct elimination), characterised by intensive bouts performed in short periods of time, with irregular breaks in between. 15 Thus, a competition can last an entire day and it is difficult to keep both physical and psychological energies under control; such a delicate aspect has been examined in several studies, [15][16][17][18][19][20] highlighting the unique cognitive processes enhanced by fencing. 21 For our purposes, we point out that the organization of tournaments itself exposes fencers to the influence of random events. Finally, at the end of a fixed competition, each placement in the final classification corresponds to a certain number of points, which scales non-linearly from top to bottom. The total points earned become part of the so-called ranking, which collect all results of every athlete who participates in at least one event during the current season. 13 The following season, each corresponding result of the previous year in the ranking is deleted and consequently updated by the new one. This updating rule probably gives more importance to a single outstanding result, less regarding a constant and pretty decent career during the years.
We discuss an agent-based model for young fencer progression and compare the results of the simulations with real data, in both official ranking and single tournaments. Our dataset consists of 100 tournament classifications of Junior Men and Women World Cups across thirteen years. Out of them, eight are supported by their respective seasonal ranking. We observe how results of a specific competition cannot be representative of athlete's talent, either their corresponding points, while their sum in the official ranking provide a better understanding of talent distribution, but only if we assume that all competitions have the same weight. In this case, we find a good agreement between simulation and data in reproducing the power-law scale of total points for Junior Men and Women (namely, fencers under 20 years old), showing a general decreasing distribution of talents in simulations, with small local fluctuations. The lognormal trend 22,23 shows that only a few people collect most of the prizes, leaving other opponents the crumbs at the end of the season. Yet, single tournament results remain unpredictable and, occasionally, surprising.
We conclude that the role of chance is significant, affecting any single point in a non-linear way. One point at a time, it influences even the final winner, thus creating a disparity between two fencers whose talent is very close in principle.
The paper is organized as follows: in Model we present our model, which is then applied to our case study in Results and Discussion; finally, in Methods we explain fencing sport more in detail and add information about our dataset.

Model
In this section we present an agent-based model realized in NetLogo environment 24 in order to reproduce the dynamics of several international competitive seasons in fencing.
In section Fencing rules of Methods we introduce the main notions of the chosen fencing discipline (épée), whose combat features make it particularly suitable for our main goal, that is the evaluation of the relative role of talent and chance in determining successful careers of athletes (fencers) belonging to a certain community.
To this aim, we need to include in each simulation run a given number N S of seasons/years, each made of a certain number N T of tournaments/events. At the beginning of each run, all the agents are randomly listed in an initial ranking. Then, every season, each athlete of the community can "choose" the number of events (≤ N T ) he wants to participate in during that year, with a probability related to the ranking order updated at the end of the previous season. Thus, each tournament is characterized by a different number N of participants.
Let us now describe how we model each single tournament, following the rules explained in Fencing rules. The structure of a standard fencing event includes two subsequent phases: the round of pools and the direct elimination table.

Round of pools
For sake of simplicity, in our model every pool is built with 6 athletes, on the basis of the scheme reported in Table 2. Thus there will be a total of N/6 pools (N should be a multiple of 6). Inside a certain pool, a given competitor fences, in turn, with each one of all the other 5 competitors into single matches (bouts) up to five touches within three minutes, keeping track of the victories, of the hits scored and of the hits received.
To decide the outcome of a given match, we associate to each of the two fencers the following corresponding probability of performing a valid touch in each subsequent time interval (varying from 2 to 60 seconds): Such a probability depends, for each fencer, both on his/her talent T k and on the chance parameter L k . In analogy with 2, 3 , we represent the talent with a real variable T k ∈ (0, 1], randomly extracted from a truncated Gaussian distribution with mean µ = 0.6 and standard deviation σ = 0.1, which includes all the qualities of the athlete (intelligence, skills, ability, training, motivation, etc.). Being an intrinsic feature of each agent, we assume that talent remains constant during an entire simulation run (made of several seasons). On the other hand, the chance parameter L k is randomly extracted for each single touch in the interval [L k − 0.3,L k + 0.3]. The mean luck factorL k characterizes the average performance of the corresponding athlete during a tournament, thus is randomly extracted in the interval [0.3, 0.7] at the beginning of each competition. The common parameter a ∈ [0, 1] in Equation (1) represents the so called talent strength, i.e. the weight of talent in making the hit; as a consequence, (1 − a) weights the importance of chance. If P 1 > P 2 we assign a valid touch to the first fencer and his/her score is increased of 1; if P 2 > P 1 the opposite happens.
It is worth to notice that the talent strength a will be the only free global parameter in our model, which will allow us to estimate -through the comparison with real data -the relative importance of talent and luck in the fencing discipline.
The possibility of double-hits during each match is also implemented with a probability calibrated on our fencing experience 1 given by: Actually, P d is the result of two different contributions of equal weight: (1 − T ), where T represents the mean talent of the two opponents; 1 −r N , beingr = |r 1 − r 2 | the difference between the initial ranking of the fencers considered. Thus, such a probability increases when both mean talent and ranking difference decrease, but it is confined in the interval [0, 0.4] thanks to the prefactor in Equation (2). For each double-hit both the competitors increase their scores of 1.
In case of tie at the end of the three minutes, an extra minute of priority is given, as already explained in Fencing rules. In the model, priority is implemented as follows: with a coin flip, priority is assigned to one of the two opponents; then, the extra minute starts and only a single touch is allowed; if a probability of a double hits is extracted the score is not updated; if the minute ends without any single valid hit, whoever owns priority wins the bout.
When all the pools are completed, a summary classification is established on the basis of several indices described in Fencing rules; as a consequence, the first 70% of the athletes after the round of pools can access to the direct elimination table.

Direct elimination table
Direct elimination table is built according to the classification after pools and can be complete or incomplete: in the former case, the number of competitors is an exact power of 2 and all bouts of that round must be held; in the latter, the number of athletes, equal to the vacancies in the table, can advance without facing any opponent.
During this phase, the bout has its canonical structure, three periods of three minutes each and a maximum score of 15 points, each touch being assigned with the same procedure implemented for the bouts in the round of pools (Equations (1) and (2)). Again, in case of a tie, an extra minute of priority is given, as already explained in Round of pools. The fencer who wins the bout advances in the table, while the loser ends his/her competition. This selective mechanism is the same in every round of the table and, at the end of the tournament, produces a pyramidal arrangement similar to that one shown in Figure 1: at the top level we found the first and the second classified; at the bottom level, the 30% of athletes who did not passed the round of pool; all the other fencers lie in the middle levels. Each agent is labeled with his/her position in the initial ranking: in the example considered, the winner of the competition started from the first position in the ranking, while the second classified started from position 7th , and so on.
At the end of the tournament, fencers receive an amount of points according to their classification, following Table 3 of Fencing rules. Notice that we do not distinguish between Championships and World Cups in our model, thus all events weight equally in simulation rankings. As explained more in detail in Fencing rules, ranking is not cumulative over the years. On the contrary, it rolls during a new season: each new result cancels out the result obtained in the corresponding competition of the previous year. The simulation stops when the last tournament of the last year ends. At this point it is possible to look at several output parameters, such as the final ranking of athletes, the correlations between their initial and final placement calculated either for each season or for each single competition and even the correlation between talent and rank, all of them as function of the selected value of the talent strength a.

Results and Discussion
We showed in the previous section that our model is able to simulate in detail the careers of athletes belonging to a given community for several seasons, each including a given set of tournaments. Our aim is that of investigating the fencing dynamics for different values of the global talent strength, and comparing the simulation results with real data in order to evaluate the relative role of talent and luck in this sporting discipline.
As explained in section Presentation of the dataset of Methods, for this comparison we chose a dataset containing the official rankings of both Junior Men and Junior Women from 2011 to 2019, with participants coming from all the countries of the International Fencing Federation (FIE). Since fencers who can access to Junior competitions must be between 14 and 20 years old, the longest possible career lasts six years. Therefore, we set to N S = 7 the number of seasons/years to simulate, considering the first as a trial stage for the following ones. According with the average length of the official FIE rankings, we consider a community of N M = 600 fencers to simulate the Junior Men seasons and a smaller community of N F = 500 fencers for the Junior Women ones.
For every season, following effective data results considered in official ranking, we simulate N T = 8 distinct tournaments with a variable number N of participants. In fact, during a given season, each athlete of the community can "choose" the number of events (≤ N T ) he wants to attend, with a probability related to the ranking order for that year. Those conditional probabilities have been extracted from the real dataset for both men and women, as explained in Presentation of the dataset (see Figure 11), and the model has been calibrated accordingly.
In Table 1 we summarize the setup of the fixed parameters or intervals that we will adopt in our simulations, included mean and standard deviation of the normal distribution of athletes' talent introduced in the previous section. In order to have statistically significant results, we will always average the outputs over 10 simulation runs, each starting from a different realization of the talent distribution among agents.

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Our goal is to find the optimal value of the talent strength a able to produce the best agreement between simulations and real data. To do so, the main observable that we decided to consider is the probability of improving, maintaining or worsening, at the end of a given season/year, the ranking placement obtained the previous year. In fact, if the official ranking was a perfect mirror of athletes' talent, those probabilities should be peaked in the corresponding placements, season after season, with very small fluctuations. Instead, observing the real data analysis results in panels (a) of both Figure 2, for men, and Figure 3, for women, there is only a weak correlation between previous and following ranking positions: for example, athletes who conclude a certain season in the first 16 positions in the ranking, at the beginning of that season were in the same first 16 positions only with a probability between 0.3 and 0.4, while are slightly less likely to have started from lower positions in the ranking, and have still a not negligible probability to have started below position 500 th . This effect is even more pronounced for the other positions, thus suggesting that talent explains only a part of the story: evidently, the influence of external factors cannot be neglected and a certain role of chance should be also taken into account.
In order to quantitatively estimate this role, we report in the other panels of the same figures the analogous results obtained with our simulations for different values of the talent strength a. It is immediate to see that the simulation outputs yielding the best agreement (inside the error bars) with real data are those obtained in correspondence of a = 0.55 for men and a = 0.45, as shown in panels (b). On the contrary, results observed for smaller (a = 0.2) or greater (a = 1) values of the talent strength, reported in panels (c) and (d) of the figures, are clearly not compatible with real data. This confirms that the role of chance in fencing competitions is quite consistent, slightly lower than 50% for men and slightly higher than 50% for women. Such a gender difference sounds realistic and could be explained by considering that girls may be typically more exposed to emotional or physiological external factors.
Once found the correct values of the talent strength, we can fix them for both men and women and go forward with other comparisons between data and simulations.
First, let us check that the conditional probabilities of having attended a certain number of competitions for simulated fencers of the Junior categories, given the associated ranking placement in the same year, are consistent with the data driven ones previously adopted for the calibration of the model. Looking at Figure 4, we can ascertain such a correspondences for men, comparing panels (a) and (b), and for women, comparing panels (c) and (d). Moreover, simulation results give back a variable number of participants to the tournaments during a given season that range from N = 186 to N = 276 for men and from N = 156 to N = 216 for women, both consistent with real data.
A very good agreement between data analysis and simulations can be registered also reporting the normalized points cumulated by the athletes at the end of the season as function of their final ranking. Plotting this information in Figure 5 by averaging over all the considered seasons for both men and women, we notice very similar trends the can be well fitted with Lognormal curves. In order to perform the fit, we used a Levenberg-Marquardt algorithm 25 , based on the following function: Comparing the data fits in panels (a) and (c) with the corresponding simulation fits in panels (b) and (d), we found fitting parameters b and c that are almost coincident.
It is also interesting to study the correlations between initial ranking and final placement in single tournaments, comparing simulation results (averaged over 80 events for both men and women) with data analysis (averaged over 52 events for men and 48 events for women). In particular, we decided to monitor the top sixteen agents in the ranking or in the final classification. For these athletes, we ask the following two questions: (1) What is the conditional probability of obtaining a certain final placement in the tournament provided that one starts from a certain initial ranking position?
(2) What is the conditional probability of having started from a certain position provided that one reaches a certain final placement in the tournament?
The answers to both the questions for male fencers can be found in Figure 6, where the (normalized) density kernel plots 26 are reported for both real data and simulated ones.
(1) The comparison between panels (a) and (b) shows that our model is able to capture very well the fact that the first 16 athletes in the ranking have a quite high probability to reach a final placement included in the first 25 positions, but -evidently due to the consistent role of chance -have also a decreasing, not negligible, probability to close the tournament in lower (and sometimes much lower) positions.
(2) At the same time, comparing panels (c) and (d), the simulations results reproduce quite well an analogous effect observed in real data, where athletes placed in the first 16 positions at the end of a tournament came from the first 30 or 40 positions in the initial ranking, but with a not negligible decreasing probability to come also from lower positions.

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(a) Data      Similar answers can be found in Figure 7, where analogous effects have been observed also for female fencers: again, a very good agreement between real data -panels (a) and (c) -and simulations -panels (b) and (d) -can be appreciated for the correlation among initial rankings and final placements.
One last consideration about talent. Of course, from the analysis of real data it is not possible to extract the distribution of talent among fencers, being the latter an hidden variable that usually is deduced by the final ranking: athletes in the top ranking positions are assumed, by definition, to be the most talented and, vice-versa, those in the bottom positions are assumed to be the less gifted. But the latter is exactly the kind of assumption which, in general, should not be done in contexts where success is heavily influenced by chance.
However, in sporting disciplines subject to competitive selection, we expect that also in presence of a not negligible role of randomness the correlation between ranking and talent would be, in some way, preserved. Simulations help us to confirm such an intuition, since we assign a fixed talent to all the fencers at the beginning of each simulation run, thus we are able to report talent as function of the final ranking. This is exactly what we do in Figure 8 for both men (a) and women (b).
As expected, we actually observe an initially decreasing trend in talent, but with increasing fluctuations around the average values as ranking get worse. Moreover, in the tail the curve starts to slowly increase again, as if certain pretty talented agents attend very few competitions during the seasons. In real data, they could represent new fencers at the beginning of their career, maybe the youngest ones, who start competing without any ranking points even if they are as talented as other older participants.
Summarizing, we can say that our agent-based model, calibrated on data of real tournaments, has allowed us to estimate the relative weight of chance (external factors, good or bad luck) with respect to talent in fencing competitions: actually, we found that this weight is very important, reaching 45% for men and 55% women! Following the line of reasoning expressed in the conclusive remarks of Ref. 3 , we could probably claim that the peculiar rules of fencing make this discipline the best candidate to provide the upper limits of chance contribution in sport competitions with individual scores, whereas the Olympic 100-meter dash studied in Ref. 3 , with its 4% for men and 6% for women, would give the lower limits. Thus, in an ideal spectrum where these two disciplines represent the extremes, all the other analogous sports -high jump, long jump, tennis, golf, car racing, motorcycle racing, etc. -would fall in the middle. If this were true, it would seriously question the methods of awarding   cash prizes which, typically, follow an exponentially decreasing trend going from the first classified to the last one. These methods are based on the implicit assumption of a one-to-one correspondence between talent of athletes and their performance in competitions. But we showed that such a correspondence does not exists, since in all these sports chance and randomness heavily influence the performance of any athlete. Let us close the paper with some economics considerations, which will be extended in future works. Sport competitions are innately motivated to foster physical improvement of mankind and establish frontiers in the ability against natural forces or among athletes. Difficulties often descend from the designed complexity of the discipline task, tremendous effort needed to reach the topmost levels, and compliance.
Every person needs great dedication and self-control to become a good athlete in any discipline, but the economic conditions surrounding the individual talent at work in different sports may create incentives in very differentiated ways. In many countries, for example, the military services sponsor young athletes, often in "non-military" sports. This provides economic support to young people, but at the same time serves as effective PR for the military services. This is frequent for the vast majority of disciplines without market-related environments. On the contrary, in other cases, rewards for winners do not seem to be conceived to remunerate the superiority of an athlete against others and, instead, they appear to serve for the creation of the highest incentive to compete, which may induce fierce competitiveness while inducing the most spectacular entertainment. This aspect is particularly interesting, from a first point of view, in order to focus on the nature of involvement in sports, specifically when comparing more or less "profitable" disciplines.
The lack of sponsors, a reduced number of fans, a smaller involvement of press, advertising and notoriety effects, all reduce the amount of money related to sport careers and championships. A football player can earn more than a tennis player who, in turn, earns much more than an archer or a fencer, despite -for example -fencing is the most medal discipline among Italian ones.
A second aspect worth noticing is related to the proportionality of differences in rewards with respect to corresponding differences in sport rankings. The possibility to measure the true impact of personal talent in sport disciplines is essential to discover the part of the success of an athlete that should be remunerated, i.e., the part measuring the "size" of the merit. Most likely, tastes of the audience cannot be guided by equative reasons, thus it is highly improbable that the Champions League final can have the same public of the final of a fencing tournament. Nonetheless, by analysing sport competitions for different disciplines, we are looking to focus on the specific components of those sports, seeking for the role of chance.
Once a reasonable range will be established, at least for individual sports, from the most aleatory ones -where randomness counts as much as (or even more) than the athlete's talent in reaching victory -to the most genuine ones -where individual talent does it almost all -, a proportionate reward scheme can be designed. In turn, this will open possible policy implications related to financing conditions, in favour of the widest opportunities for athletes in all disciplines. The economic exploitation of market-oriented implications of sport competitions should not cause distortionary effects against pluralism of disciplines. "Profitable" disciplines could be helpful for other ones, e.g., in helping young athletes starting their career, by draining resources in excess from the formers to the latter.

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Methods In this section we describe fencing technical rules, competition formula and point assignment in ranking during an entire season. 2 Fencing is a combat sport involving a weapon, which can be of three different kinds, identifying three separate disciplines: foil, sabre, épée.

Fencing rules
Each weapon has its own peculiarities, but they also have common characteristics: the two opponents compete on a piste, 14 metres long and 1.5 metres wide; the goal is to score a valid touch on your opponent, which counts as a point; the first fencer that achieves 15 points, in a bout (individual match) composed at most of three rounds (called periods) of three minutes each, wins; touches and time are controlled by a referee, according to an electrical recording apparatus.
In case of tie at the end of the third period, one extra minute is given, assigning a priority at random to one fencer: if no one scores a single touch during this time, the opponent with priority wins. Therefore, the role of randomness is explicitly present in fencing rules, making this sport an interesting candidate in studying the effects of chance (good luck, bad luck or other external factors) in competitions.
In this work, we focus on épée because there are no right of way rules regarding attacks, which means that any hit is counted and the point is assigned to the fencer who makes the hit first. As a consequence, referee discretion is strongly limited and we can neglect human error contribution.
The main features of épée are the following: • the attack is possible only with the point of the weapon; • the target area is the entire body; • since any touch is counted, double-hits are allowed if they occur within 40 milliseconds.
Those very essential rules allow épée tournaments to be simulated by an agent-based model, as described in Model. But first, we need to explain the organization of fencing competitions.
Usually, a competition (tournament) consists of two main phases, one round of eliminating pools ( Table 2) followed by a direct elimination table (Figure 9). The pools comprise 6 or 7 fencers, depending on the number of participants. They are composed taking into account the latest official FIE ranking, which collects the points obtained in the previous events of the current season or in the previous season on the basis of athletes' placements.
The allocation of fencers in the pools follows the method shown in Table 2. Table 2. Allocation method explained for three different pools (A, B and C), based on ranking placements of 18 competitors. Each column represents a complete pool with 6 fencers.
In pool rounds, each competitor fences a bout against all of the other members of their pool, up to five hits in only one period of three minutes and an extra priority minute in case of tie.
After the pools, a single general ranking of all the athletes is established, on the basis of the following indices: first, V M is considered, where V = is the number of victories and M = the number of bouts; then, in case of equality, the difference HS − HR between the hits scored (HS) and the hits received (HR) is taken into account; finally, in case of further equality in both V M and HS − HR, the fencer who has scored most hits (highest HS) is seeded highest. In the special case of absolute equality, the order is decided by drawing lots.
From the round of pools, only 70% of the fencers is qualified for the direct elimination phase, depending on the classification after pools.
Direct elimination table consists of many rounds that scale with decreasing powers of 2 (usually 256, 128, 64, 32, 16, 8, 4 and final) as shown in Figure 9: as we can see, the first classified after pools is coupled with the 64 th , the second one with the 63 rd and so on.
For example, if the competition starts with 100 participants, after pools there are 70 athletes qualified to the direct elimination round, which is an incomplete table of 128 (since they are more than 64). Therefore, the last 12 athletes, from number 59 up to 70 after pools, have to win one more match to access the table of 64, while the first 58 fencers automatically advance. In detail, the couples are: 59-70, 60-69, 61-68, 62-67, 63-66, 64-65. Once these matches are completed, table of 64 starts for all the participants left, following the bouts indicated in Figure 9. Intuitively, in each round the fencer who wins his bout have access to the next one, while the loser ends his competition and obtains a placement coincident with the reached round, according to the ranking after pools. For example, if fencer A loses against fencer B in the table of 16, fencer A can be placed in the classification among the 9 th and the 16 th place.
The general classification is compiled from the winner of the final bout, who is also the winner of the competition, followed by the second, the one who loses the bout for the first place; then, there is an ex aequo for the third place, assigned to the two fencers defeated at round of semi-finals; the other placements are given as explained above.
The goal of attending competitions is to rise in the official FIE ranking. In fact, at the end of each tournament, all the

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participants gain a certain amount of points, fixed by the following scale:  Table 3. Scale points in the official FIE ranking.
We can see that the scale decreases in a non-linear way, following a power law from the 9 th place on. Notice that some values were added later: * was introduced in season 2015/2016; * * were introduced only in 2019.
Ranking is not cumulative over years, instead it rolls during the season: the new result cancels out the previous year result in the corresponding competition. Moreover, the official Junior ranking of the FIE considers only the best six results of the World Cup events in which the fencer has participated, other than the Zonal and World Championships, for a total of eight results collected. As a consequence, only those athletes who actually attended at least one event in a season are listed in the ranking.
We need to precise that different kinds of competitions weight differently in ranking: points obtained in World Cups events are multiplied by a factor of 1; Zonal Championship points in our dataset are multiplied by a factor of 1.5 (this rule has been updated in season 2019/2020, the factor being reduced to 1); points obtained in World Championships are multiplied by a factor of 2.5.

Presentation of the dataset
In the previous subsection we have introduced the main rules of fencing, with particular regard to épée discipline, as well as the competition formula adopted by the type of events which we are interested in. Those ingredients are all important to understand the construction of the official ranking, whose analysis is our main purpose.
We collected the official Junior Men and Women rankings from 2011 to 2019, excluding previous years because there were very different criteria in point assignment and the year 2020 since it is incomplete (World Championship has been cancelled due to COVID-19 pandemic). All participants are in between 14 and 20 years old and they are from all the countries that are FIE members. The official rankings in our dataset come from FIE website 13 . Since the longest possible career in Junior category lasts six years (as already mentioned in Results and Discussion and we have nine years available, the group of athletes involved is not constant. Instead, every season there are some fencers who become too "old" and must change their category; on the other hand, "young" fencers who can participate in Junior Events for the first time come into play. For this reason, official ranking does not have a fixed number of competitors; we find that the average length for Junior Men is about 600 participants, while for women is about 500 in our dataset, as reported in Results and Discussion. We decide to analyse total points as a function of ranking order, highlighting their non-linear trend by fitting them with a lognormal function (see Equation (3) and Figure 5 in Results and Discussion). Notice that total points are averaged over the seasons. The results of the fit are summarised in Table 4.  Table 4. Summary of the results for the non linear fit of ranking points, using Equation (3), both for data and simulations with the chosen talent strength a = 0.55 for men and a = 0.45 for women.
We also collected the initial rankings and final classifications of 100 World Cups, from 2008 to 2020, to perform the analysis of single tournaments shown in Results and Discussion. Data availability was an issue in this case, several sources were used in 15/19 the attempt of extending the dataset as much as possible. 13,[27][28][29][30][31][32][33] In details, our data comprise 52 results for Junior Men and 48 for Junior Women, considering only events with at least 100 competitors. We observed that the number of participants in a competition can reach a maximum value of 280 for men and 230 for women, with an average around 170 and 150 respectively, as visible in Figure 10. From the variety of participation in tournaments, it follows that not even the list of participants in single competitions can have a fixed length. That is why we let the number of participants N vary in simulation runs (see Results and Discussion). It is quite natural that athletes attend only a certain number of events during a given season, according to their position in the ranking and to their country's instructions. One should think that the higher the position in ranking, the more events the athlete took part in. Data analysis for both men and women shows that this is generically correct, but we find a non-trivial probability distribution of ranking placements conditional to the actual attended events (see Figure 11) which allowed us to calibrate the model.  If we focus on the first sixteen athletes, who are usually considered the top performers, we can look for correlations between ranking positions before a certain competition and the final results of the same event, and viceversa, for both men and women. We evaluate the median value of the initial/final ranking for the top sixteen fencers, rather than the mean value, because of the asymmetric nature of data themselves. In Figure 12 we show the results. The shadows indicate the median absolute deviation, a measure of spread which is suitable for median values.    In the upper panels men and women data are fitted with linear functions, while in the lower panels are fitted with a non linear (quadratic) function. In Table 5 the fitting parameters for both the functions are reported with the corresponding errors.

Data
Linear Fit y = q + mx Non Linear Fit y = (  Table 5. Parameters and residual standard error (RSE) of the linear and non linear fit, which are modelled over the median values of initial versus final rankings in single tournaments and viceversa.
We precise that the statistical analysis was performed using R software 34 .