From cut-off to kick-off: Dynamic modeling and mitigating of the relative age effect

Abstract

This study introduces a dynamical model to analyze and mitigate the Relative Age Effect (RAE) among male handball players in the French federation, from youth to professional levels. The RAE arises from age-based grouping, leading to psychological and behavioural (motor) differences among young athletes. Using system dynamics, the model simulates player movement and cumulative experience across cohorts, calibrated with data from the Fédération Française de Handball. The model incorporates a reinforcing feedback loop that amplifies initial advantages in size and weight, enhancing perceived skills and training opportunities. Various policy scenarios are explored to assess their impact on reducing the RAE. Results suggest that modifying cut-off dates could be the most effective strategy, potentially improving talent detection and reducing attrition in young cohorts. This research offers valuable insights for decision-makers aiming to address the RAE in handball and other team sports.

Keywords

Handball sport attrition training capacity youth development

Introduction

When we think of success, we consider a mix of several factors such as hard work, talent and luck. Duckworth et al.¹ introduce the concept of “grit,” which is a combination of perseverance and passion for long-term goals. This concept applies to various fields, including economics, education or sports and more broadly, to many aspects of life. If we look more precisely at these factors on the scale of a year or a cohort, we can add another factor: the Relative Age Effect (RAE). Relative Age is the difference between an individual’s birth date and the cutoff date established for participation in a specific sport or age-group activity. The effect is a phenomenon that describes the advantage or disadvantage some individuals may experience based on their birth date. In sports, young players are grouped by birth date, which can result in an age difference of almost one year between two players in the same category. In young cohorts, this age gap can lead to significant differences in players’ psychological development and motor skills. The Relative Age Effect (RAE) is well documented in many sports,² such as soccer,³ hockey,⁴ rugby,² basketball,⁵ handball⁶ and many others. Hancock et al.⁷ and Wattie et al.⁸ propose a developmental systems model to better understand the Relative Age Effect (RAE), particularly in its multifactorial nature. They show that the Relative Age Effect (RAE) is a more complex phenomenon than just physical or psychological advantages, as it results from intricate interactions between various factors, including biological, psychological, social, and environmental influences. Hancock et al.⁷ base her model on three sociological and psychological theories. First, the Matthew Effect, which explains that an initial advantage can lead to increasing advantages over time. Second, the Pygmalion Effect, which suggests that the expectations of others can influence an individual’s performance. Third, the Galatea Effect, which states that an individual’s own expectations about their abilities can impact their performance.

A powerful tool for modeling complex systems that evolve over time is dynamical systems theory. The model proposed by Pierson et al.⁹ introduces the first behavioral dynamic framework to explain the Relative Age Effect (RAE) in hockey through a single positive feedback loop. The most effective policy combines rotating cutoff dates (Jan/Jul) and targeted support for younger-born players, reducing the RAE by 96% at equilibrium, though effects take 20–30 years to materialize at the professional level. Our model extends this approach by incorporating additional feedback structures, real federation data, and more complex learning and policy mechanisms to better reflect practical decision-making within the French Handball Federation.

The main objective of this article is to create a dynamical model to describe the complex mechanism of the Relative Age Effect (RAE) from the youngest cohorts to the professional level in handball, specifically within the French federation and among male players. This article is structured as follows. First, we briefly introduce the concept of dynamical systems and present the data used in our study. Then, we construct the dynamical model and use the data to calibrate it. Finally, we apply the model to test different policy scenarios in order to understand their effects on the Relative Age Effect (RAE).

Material and methods

System dynamics

The system dynamics method is a powerful approach for modeling and understanding complex real-world issues. It is used in various fields, such as economics,¹⁰ epidemiology,¹¹ ecology,¹² and, by extension, in domains that involve complex situations and nonlinear equations. System dynamics is built on some basic structures: stocks, flows, and loops. Stocks represent the accumulations or the state of the system at any given point in time. They function as “containers” that hold resources, quantities, or information. Flows represent the rates at which stocks increase or decrease. They act as the “pipes” or “arrows” that transfer resources or quantities into or out of stocks. There are two types of flows: inflows and outflows. Loops illustrate the interactions and feedback within a system. They explain how changes in one part of the system influence other parts and create cycles of influence. There are two types of loops: reinforcing (positive) loops, which amplify changes in the system, and balancing (negative) loops, which counteract changes to stabilize the system. These elements drive the behavior of the system and can be used to model complex real-world issues effectively.¹³

Data

The data used to fit the model and estimate the parameters come from the Fédération Française de Handball (FFH). FFH uses five cohorts (U9, U11, U13, U15, U18), and professional players are added as an extra cohort in equation (20).rc1 This dataset consists of male player license registrations across all handball categories (from U9 to veterans) from 2015 to 2024. Specifically, we have collected players’ birth dates, and by consequences their categories, and their longitudinal trajectories throughout their handball careers. We can use this data to estimate, for a given cohort, the rate of players who permanently leave handball, the rate of players entering from external sources, and the initial proportion of players in each cohort. We chose to aggregate the data by birth quarters rather than birth months to mitigate large variations observed over the years 2015 to 2024. Table 1 shows the distribution of birth quarters for professional players (sourced from the FFH) and the general population (sourced from Institut National de la Statistique et des Etudes Economique (INSEE)).

Table 1.

Distribution of birth quarters among professional players (data source: FFH), the general population (data source: INSEE), and model predictions, expressed as percentages (%).

Quarter	Q1		Q2		Q3		Q4
INSEE	23.7		24.7		26.3		25.3
Year	Observed	Modelled	Observed	Modelled	Observed	Modelled	Observed	Modelled
2015	30.15	31.62	27.45	24.87	22.79	24.15	19.61	19.36
2016	29.83	31.37	27.87	25.05	22.25	24.20	20.05	19.38
2017	30.99	31.13	24.41	25.19	25.12	24.24	19.48	19.44
2018	32.23	30.93	26.45	25.31	21.69	24.26	19.63	19.51
2019	33.47	30.77	25.96	25.41	23.12	24.26	17.44	19.56
2020	31.52	30.66	23.64	25.52	24.65	24.25	20.20	19.57
2021	34.08	30.61	22.32	25.64	24.74	24.22	18.86	19.53
2022	32.82	30.61	23.62	25.79	24.23	24.17	19.33	19.44
2023	29.68	30.64	24.13	25.95	26.51	24.10	19.68	19.31
2024	30.69	30.69	24.83	26.12	25.00	24.01	19.48	19.17

Model structure

Coflow model

The main mechanism of the model structure is a reinforcing feedback loop that causes the Relative Age Effect (RAE) in young sports (see Figure 1). Players enter as part of a cohort with a specific relative age, following the distribution of birth quarters in the global population. This can be view as Matthew effect.⁷ Differences in size, weight, psychological development, or motor skills can lead to variations in how trainers perceive players’abilities. As a result, they tend to select early-born players more often than others, providing these players with more training and playing opportunities. Consequently, these players gain more relative experience. Over time, this additional experience leads to better perceived skills by the trainers. The Relative Age Effect (RAE) becomes a self-reinforcing loop that amplifies the initial advantages in weight and size, compounding across age cohorts. This effect is well documented, for example, in Musch and Grondin,¹⁴ Cobley et al.,² Bedard and Dhuey,¹⁵ Wattie et al.¹⁶ or Hancock et al.¹⁷

Figure 1.

The positive feedback loop of the Relative Age effect from Pierson et al.⁹

We use a coflow model, composed by two principal age-chain equations. Let $1 \leq q \leq 4$ the quarter index, $1 \leq c \leq 5$ the cohort index. The first equation simulated the movement of the players in each cohort and for each quarter and the second one simulated the cumulative experience acquired in each cohort and for each quarter.

\frac{d P_{q c} (t)}{d t} = I_{q c} (t) + I_{q c}^{e} (t) - E_{q c} (t) - Q_{q c} (t)

(1)

\begin{aligned} \frac{d C E_{q c} (t)}{d t} & = I E_{q c} (t) + I E_{q c}^{e} (t) + T E_{q c} (t) - E E_{q c} (t) \\ - Q E_{q c} (t) \end{aligned}

(2)

In equation (1), $P_{q c}$ represents the number of players in cohort $c$ during quarter $q$ . $I_{q c}$ is the inflow of players entering the cohort from a lower development stage, while $I_{q c}^{e}$ represents the inflow of players joining the cohort who do not belong to any lower development stage. $E_{q c}$ denotes the outflow of players exiting the cohort to progress to a higher development stage, and $Q_{q c}$ corresponds to the outflow of players leaving both the cohort and handball entirely.

The inflow of cohort $q$ is determined by the outflow of the preceding cohort $c - 1$ , i.e., $I_{q c} (t) = E_{q (c - 1)} (t)$ .

The external inflow follows the same process as the first inflow, as it consists of players originating from the general population. Their numbers are proportional to the birth rate in France and the size of the cohort, adjusted by the external inflow fraction $I F_{e x t}$ . $τ_{c}$ represents the delay within each cohort and $I F_{q}$ is the inflow of the youngest cohort.

I_{q c}^{e} (t) = \frac{I F_{q} \cdot I F_{q c}^{e x t} \cdot \sum_{q = 1}^{4} P_{q c} (t)}{τ_{c}}

(3)

The total outflow of a cohort is proportional to the time delay $τ_{c}$ and the size of the cohort.

E_{q c} (t) = \frac{P_{q c} (t)}{τ_{c}} \cdot (1 - Q F_{q c})

(4)

Q_{q c} (t) = \frac{P_{q c} (t)}{τ_{c}} \cdot Q F_{q c}

(5)

where

Q F_{q c}

is the quit fraction for month

q

and cohort

c

, as defined later in equation (17).

In equation (2), $C E_{q c} (t)$ represents the cumulative experience of all players in cohort $c$ during quarter $q$ . $I E_{q c} (t)$ corresponds to the experience gained by the inflow of players from a lower state. $I E_{q c}^{e} (t)$ represents the experience of the external inflow of players, and, as previously stated, it is equal to the exit experience of cohort $c - 1$ , i.e., $E I_{q c} (t) = E E_{q (c - 1)} (t)$ .

Let $A v g E_{q c} = \frac{C E_{q c}}{P_{q c}}$ be the average cumulative experience for quarter $q$ and cohort $c$ ,

I E_{q c}^{e} (t) = I_{q c}^{e} (t) \cdot C o E_{c} \cdot min (A v g E_{q c})

(6)

where

C o E_{c}

is the experience acquired by an exterior player.

Exit $E E_{q c} (t)$ and quit experience $Q E_{q c} (t)$ are proportional to the average cumulative experience of cohort $c$ during quarter $q$ .

E E_{q c} (t) = A v g E_{q c} \cdot E_{q c}

(7)

Q E_{q c} (t) = A v g E_{q c} \cdot Q_{q c}

(8)

Physical training and reinforcing loop

Skills and experience can be linked through different types of curves.¹⁸ Common learning curves include Wright’s model,¹⁹ the Plateau model, the Stanford-B model,²⁰ DeJong’s model,²¹ and the S-curve model.²² For this study, we chose a different model to link skills and experience: the hyperbolastic rate equation of type II (H2),²³ defined as

\frac{d P (x)}{d x} = \frac{α δ γ P^{2} (x) x^{γ - 1}}{M} \tanh (\frac{M - P (x)}{α P (x)})

(9)

where

\tanh

denotes the hyperbolic tangent function,

M

represents the carrying capacity, and the parameters

δ

and

γ > 0

together influence the growth rate of a population of size

P (x)

. The parameter

γ

specifically indicates the rate of acceleration over time. These models were initially developed to describe the growth dynamics of multicellular tumor spheres.²⁴

Solving this equation leads to

P (x) = \frac{M}{1 + α arsinh (e^{- δ x^{γ}})}

(10)

Skill $S_{q c}$ is linked to experience $C E_{q c}$ as follows:

S_{q c} = \frac{100}{1 + α arsinh (\exp (- δ {(A v g E_{q c})}^{γ}))}

(11)

The parameters

α

δ

, and

γ

must be estimated during the model calibration process. The relationship between skills and training is modeled as described in Pierson et al.⁹ We use a logit choice model²⁵ (equation (13)).

Moreover, each cohort may exhibit different sensitivity to training, which affects the quantity of training they receive. This can be view as Pygmalion effect.⁷ To model this behavior, we introduce a variable $T S_{c}$ (Training Sensitivity), and the relationships are defined as follows:

T A_{q c} = \frac{S_{q c}}{\sum_{q = 1}^{4} S_{q c}}

(12)

T F_{q c} = \frac{\exp (T S_{c} \cdot \log (T A_{q c}))}{\sum_{q = 1}^{4} \exp (T S_{c} \cdot \log (T A_{q c}))} = \frac{{(T A_{q c})}^{T S_{c}}}{\sum_{q = 1}^{4} {(T A_{q c})}^{T S_{c}}}

(13)

T E_{q c} = P_{q c} \cdot (C o T_{c} + (T F_{q c} - \frac{1}{4}) \cdot C o T_{c}^{q a x})

(14)

Here $T A_{q c}$ represents the training affinity for quarter $q$ and cohort $c$ , $T F_{q c}$ is the training fraction computed using the choice model, and $T E_{q c}$ is the training experience gained from training. $C o T_{c}$ denotes the annual quantity of training for a cohort.

$C o T_{c}^{m a x}$ represents the additional annual training time if a player receives full attention from their coach (e.g., through supplementary training or matches).

Back to the model

Let $A v g S_{c}$ be the average skill for cohort $c$ over the quarters, defined as:

A v g S_{c} = \frac{\sum_{q = 1}^{4} S_{q c}}{4}

(15)

and let

S G_{q c}

represent the skill gap:

S G_{q c} = \frac{A v g S_{c} - S_{q c}}{A v g S_{c}}

(16)

We can model the quit fraction $Q F_{q c}$ as follows:

Q F_{q c} = max (min (N Q F_{c} + Q S \cdot S G_{q c}, 1), 0)

(17)

Here, $N Q F_{c}$ represents the normal quit fraction for cohort $c$ . $Q S$ denotes the strength of the effect that skill gaps have on the quit fraction and must be calibrated. This formulation, taken from Pierson et al.,⁹ implies that a 1% skill deficit or advantage relative to the cohort’s average results in a $Q S$ % increase or decrease in quitting. $Q S$ can be view as Galatea effect.⁷

The complete model is illustrated in Figure 2 and a summary of the parameters is provided in Table 2.

Figure 2.

Complete diagram of the model.

Table 2.

List of the model parameters with a short description. The “Status” column indicates how the value was determined, and we specify the number of parameters concerned in each row. In total, there are 54 parameters.

Parameters	Number of parameters	Status	Description
$I F_{q}$	4	Fixed with INSEE data	Fraction of birth quarter for the very young players
$I F_{q c}^{e x t}$	20	Fixed with FFH’s data	Fraction of exterior players arriving in a cohort $c$ and quarter $q$ (e.g. without previous handball experience)
$τ_{c}$	5	Fixed with FFH’s data	Delay for a players staying in a cohort $c$
$C o E_{c}$	5	Fixed	Fraction of experience of exterior player
$M$	1	Fixed	Carrying capacity. Arbitrarily fixed to 100
$α$	1	Fitted by the model	Internal parameter of the hyperbolastic model
$δ$	1	Fitted by the model	Growth rate of the hyperbolastic model
$γ$	1	Fitted by the model	Acceleration over time of the hyperbolastic model
$T S_{c}$	5	Fitted by the model	Training sensitivity
$C o T_{c}$	5	Fixed	Annual quantity of training for a cohort $c$
$C o T_{c}^{m a x}$	5	Fixed	Additional annual training
$Q S$	1	Fitted by the model	Strength of the effect that skill gaps have on the quit fraction

Model calibration

The output of the model is the fraction of professional handball players for each birth quarter and each cohort, denoted $B F M_{q}^{p r o}$ :

B F M_{q}^{p r o} = \frac{P_{q}^{p r o}}{\sum_{q = 1}^{4} P_{q}^{p r o}}

(18)

Note that the model returns the fraction of each cohort $B F M_{q c}$ , too:

B F M_{q c} = \frac{P_{q c}}{\sum_{q = 1}^{4} P_{q c}}

(19)

The model’s parameters are fitted using the R package FME²⁶ and the ModFit function with PORT routines. The function to be minimized is

\sum_{c = 1}^{6} \sum_{q = 1}^{4} {(B F M_{q} - B F M_{q}^{t h})}^{2}

(20)

with the final objective of achieving a strong modeling of the professional proportion. We could have minimized only

\sum_{q = 1}^{4} {(B F M_{q}^{p r o} - B F M_{q}^{p r o, t h})}^{2}

(21)

but we had less data compared to using all fractions.

Scenarios

The first scenario, noted $S_{1}$ , and perhaps the most challenging to implement, involves introducing a policy that provides greater support to players born in the last quarters of the year. These measures are reflected in the model through parameters $T S_{c}$ , $C o T_{c}$ , or $C o T_{c}^{m a x}$ . We break down this scenario into several parts: $S_{11}$ corresponds to a 20% reduction of $C o T_{m a x}$ , i.e., $C o T_{m a x} = 4$ , $S_{12}$ involves a 50% reduction of $C o T_{m a x}$ , i.e., $C o T_{m a x} = 2.5$ , $S_{13}$ corresponds to a 50% reduction of all $T S_{c}$ , $S_{14}$ represents a very intensive adjustment of $T S_{c}$ , especially for young players, with $T S_{1} = T S_{2} = T S_{3} = - 1$ and $T S_{5} = T S_{6} = 0$ . $S_{15}$ is a mixed version where each $T S_{c}$ and $C o T_{m a x}$ are reduced by 20% which is realistic in practice because these parameters are linked and this correction rate can likely be reached with some actions.

The second scenario, noted $S_{2}$ , is the rotating cut-off, which consists of changing the cohort cut-off date each year. Grondin et al.²⁷ proposed a 15-month or 21-month category system instead of the traditional 12-month system, thereby breaking the structure based on multiples of 12. We have two possible approaches to implementing this system: fixed-Length Categories, which define categories based on a 15-month or 21-month period with a fixed starting month (e.g., from January to September of the following year), and rotating Cut-Off, which alternates the cut-off month each year to prevent the system from consistently favoring certain birth periods. We can alternate the cut-off date to January and April, noted $S_{21}$ , January and July, noted $S_{22}$ , January and October, noted $S_{23}$ , or we can alternate the cut-off date each year, noted $S_{24}$ . These scenarios are summarized in Table 3. These two families of scenarios are often discussed in some articles (for example, Barake et al.²⁸ or Yamamoto et al.²⁹) as solutions or recommendations for the elimination of RAE, but without a verifiable way to test this assumption.

Table 3.

Summary of all scenarios tested in the model.

Scenario	Scenario family	Main change	Description	Expected effect on RAE
$S_{11}$	$S_{1}$ – Support to last quarters	$↓ C o T_{c}^{m a x}$ by 20%	Slight reduction of the maximum annual training capacity	Small improvement
$S_{12}$	$S_{1}$ – Support to last quarters	$↓ C o T_{c}^{m a x}$ by 50%	Strong reduction of the maximum annual training capacity	Moderate improvement
$S_{13}$	$S_{1}$ – Support to last quarters	$↓ T S_{c}$ by 50%	Reduced training sensitivity for all cohorts	Moderate improvement
$S_{14}$	$S_{1}$ – Support to last quarters	Intensive $T S_{c}$ adjustment	Intensive support for youngest players ( $T S_{1} = T S_{2} = T S_{3} = - 1$ , $T S_{5} = T S_{6} = 0$ )	Strong improvement
$S_{15}$	$S_{1}$ – Support to last quarters	$↓ T S_{c}$ and $↓ C o T_{c}^{m a x}$ by 20%	Combined moderate correction on both parameters; realistic in practice	Strong and realistic improvement
$S_{21}$	$S_{2}$ – Rotating cut-off	Alternating January / April	Change of the cut-off date each year between January and April.	Slight improvement
$S_{22}$	$S_{2}$ – Rotating cut-off	Alternating January / July	Change of the cut-off date each year between January and July.	Moderate improvement
$S_{23}$	$S_{2}$ – Rotating cut-off	Alternating January / October	Change of the cut-off date each year between January and October.	Strong improvement
$S_{24}$	$S_{2}$ – Rotating cut-off	Full rotation (Jan $\to$ Apr $\to$ Jul $\to$ Oct)	Cut-off month rotates every year across all quarters	Best global improvement

To test this, we compute the sum of the squared differences between the simulated birth quarter distribution and the INSEE distribution in comparison to the base case. This allows us to obtain a percentage improvement in the Relative Age Effect (RAE).

\frac{\sum_{q = 1}^{4} (B F M_{q}^{p r o, S_{j}} - B F M_{q}^{p r o, t h})^{2} - \sum_{q = 1}^{4} (B F M_{q}^{p r o} - B F M_{q}^{p r o, t h})^{2}}{\sum_{q = 1}^{4} (B F M_{q}^{p r o} - B F M_{q}^{p r o, t h})^{2}}

(22)

where

k

and

ℓ

are integers such that

S_{k ℓ}

covers all scenarios, and

B F M_{q}^{p r o, S_{j}}

represents the proportion of professional players in quarter

q

for scenario

S_{k ℓ}

Results

Parameters

The parameters are provided in Table 4. The training affinity for cohorts U9 and U11 is set to 0. This is consistent with the fact that these young players are not influenced by additional specific training.³⁰ At this age, the emphasis is on general development and free play rather than structured training. Figure 3 shows the trajectories from 2015 to 2050 of the birth quarter fraction for professional players, with dots representing the sample data. We observe that the model fits the sample well and that the proportion stabilizes over time. Overall, the model provides a good fit to the data, with $R^{2} = 72.89 %$ , but this increases to $R^{2} = 87.86 %$ when considering only the main objective, which is the proportion of professional players.

Figure 3.

Trajectories of the quarters birth proportion for professional players. The dots represent sample data.

Table 4.

Estimated parameters from our model with 95% confidence intervals. $δ$ , $γ$ , and $α$ are parameters associated with the growth curve. $T S_{i}$ are the training sensitivity of the cohort $i$ and $Q S$ is the quitting strength.

	$δ$	$γ$	$α$	$T S_{1}$	$T S_{2}$	$T S_{3}$	$T S_{4}$	$T S_{5}$	$Q S$
Parameters	0.3229	1.2000	999.9656	0.0000	0.0000	0.1595	2.4503	0.0675	0.0882
Lower	0.0000	0.0000	999.9655	0.0000	0.0000	0.0000	2.4175	0.0672	0.0000
Upper	3.9152	4.2571	999.9658	0.2194	0.2625	0.4422	2.4831	0.0678	7.0309

Sensitivity analysis $T S$ values

We explore the effect of $T S_{c}$ on the model. Figure 4 presents a visual comparison of model runs for different values of $T S_{c}$ , while keeping other variables unchanged. The values are generated using a Latin square distribution within a plausible range from 0 to 5. We generate 200 data samples and, for each year, plot the average proportions. The dark color represents the range of trajectories within the mean plus or minus the standard deviation, while the light color shows the range between the minimum and maximum values. It appears that for certain values, the proportions tend to converge to similar percentages.

Figure 4.

Sensitivity analysis of the professional fraction for the variable training affinity. $T S_{i}$ for $i = 1, \dots, 6$ varies in the range $0$ to $5$ following a Latin square distribution, while $Q S$ is fixed at 0.1.

Testing scenario

In Figure 5, we can see the results of the model simulation. For the $S_{1}$ family, the Relative Age Effect (RAE) is always improved, and the more attention and intensive support we provide to players, the more the effect of the Relative Age Effect (RAE) diminishes. The order from worst to best is: $S_{11} < S_{13} < S_{12} < S_{15} < S_{14}$ . We can note that a reduction in the maximum training capacity is more effective in reducing the Relative Age Effect (RAE) than reducing the sensitivity of training. In both the $S_{12}$ and $S_{13}$ scenarios, we reduce the impact of these variables by 50% and the improvement is greater in $S_{13}$ . For the $S_{2}$ scenario family, the Relative Age Effect (RAE) is always improved in order $S_{21} < S_{22} < S_{23} < S_{24}$ .

Figure 5.

The percentage improvement in the Relative Age Effect (RAE) is computed as the sum of the squared differences between the simulated birth quarter distribution and the INSEE distribution, compared to the base case. The order from worst to best is: $S_{11} < S_{13} < S_{12} < S_{15} < S_{21} < S_{14} < S_{22} < S_{23} < S_{24}$ where $S_{i j}$ represents the $j$ -th scenario for the $i$ -th case. $S_{11}$ corresponds to a slight reduction in the maximum annual training capacity, $S_{12}$ to a strong reduction in the maximum annual training capacity, $S_{13}$ to a reduction of training sensitivity for all cohorts, $S_{14}$ to a very intensive support for youngest players, $S_{15}$ to a combination of moderate correction on both parameters. The $S_{2}$ scenarios involve changes to the cut-off date: $S_{21}$ shifts the cut-off each year between January and April, $S_{22}$ each year between January and July, $S_{23}$ each year between January and October, $S_{24}$ rotates every year across all quarters.

Discussion

Model justification

In the equations above, many parameters are initialized using the data from the FFH. In equation (1), the initial stock of players for each cohort was calculated using this data. In equation (3), $τ_{c}$ and $I F_{q c}^{e x t}$ are calibrated too with this data and in equation (17), $N Q F_{c}$ is a value determined using data from the FFH for each cohort.

For the first cohort, i.e., the youngest players, the inflow $I F_{q}$ (equation (1)) is set to the birth distribution in France, as reported by INSEE³¹ with the external inflow set to 0. The experience inflow of the youngest cohort (equation (2)) is designed to mimic the Relative Age Effect (RAE) by assigning players born in January 23/24ths of an additional year of experience, gradually decreasing this value to 1/12th of a year for each subsequent month.⁹ In equation (6), we assume that players from external sources lack handball-specific experience; their experience is set to the minimum experience of a player in cohort $c$ , scaled by a coefficient. This accounts for the fact that external players may improve their physical attributes and possibly acquire skills in other sports, but they do not possess specific skills in the field of handball. Therefore, the cumulative external experience is proportional to the number of external players and to a fraction of the experience gained by cohort $c$ . $C o E_{c}$ is set to 0.5 because, although the players were not playing handball, they may have participated in other sports, thereby gaining general experience rather than sport-specific experience. The expression with the hyperbolastic formulation of the learning curve is justified by the shape of the solution curve for different parameters. In Figure 6, we illustrate several solutions for different parameter values, and we observe shapes similar to those of classical models. Thus, this model allows us to have a large variety of solutions and is more flexible than others (Wright’s model, Plateau model, Stanford-B model, DeJong’s model or S-curve model). In this curve, we use actual skills, while some authors use perceived skills, which are the skills of players that coaches can observe. Pierson et al.⁹ use a first-order information delay to differentiate between actual skills and perceived skills. They note that the value of the delay does not influence the results of the simulation. Hence, we assume that coaches perceive any change in a player’s skills instantaneously, without delay. In equation (14), $C o T_{c}$ is calculated based on the time a player spends in a cohort over one year. For cohorts ranging from U9 to U18, it is set respectively to 1.6, 2.4, 3.2, 4, and 4.8. For example, in the U9 cohort, the FFH recommends approximately 2–3 sessions per week, each lasting 1.5 h, totaling approximately 160 h per year, which gives the coefficient 1.6. We do the same for the other categories. In equations (13) and (14), if a cohort has no sensitivity to training ( $T S_{c} = 0$ ), the training fraction becomes $T F_{q c} = 1 / 4$ , and training experience is distributed equally across all months. However, if a cohort has high sensitivity, differences in training affinity are amplified, and training experience is greater in certain months. $C o T_{c}^{m a x}$ is set to 5, meaning that a player can gain up to five times more experience through playing and training compared to a player who plays and trains less. Discrete choice models describe and explain the selection among two or more discrete alternatives by specifying the probability that an individual chooses one option from a set of alternatives. Training experience increases if a coach perceives a player’s skill and, consequently, provides the player with training or match opportunities. Thus, the coach either chooses or does not choose a particular month for a cohort. In this context, the conditional logit model is well-suited. In this model, the probability of selecting a specific month for a cohort is given by $P_{q c} = \frac{\exp (β z_{q c})}{\sum_{q = 1}^{4} \exp (β z_{q c})}$ where $β$ represents model parameters, and $z_{q c}$ are covariates. Thus we have the expression of $T F_{q c}$ in equation (13).

Figure 6.

Shape of 250 curves from hyperbolastic rate equation of type II for $0.0001 \leq δ \leq 1.51874$ , $0.0001 \leq γ \leq 2.35172$ and $999.82867 \leq α \leq 999.82889$ .

An interesting point regarding the fitted parameters of the model is the value of $T S_{4}$ , which is significantly different from 0. $T S_{4}$ represents the sensitivity to training of the cohort $U 15$ , composed of players aged 13 to 15 years who are entering puberty, while younger cohorts are predominantly composed of prepubescent players.³² This period is associated with rapid modifications in body proportions and composition, as well as a more anabolic hormonal environment, which could influence the response to training.³³ However, some recent studies question this model of youth development. This is not a discrepancy with our model, because the parameter $T S_{4}$ can be interpreted as an index applicable to all players in the cohort over the entire time period. It is not an individual-level parameter, but a group-level one.

Testing scenarios

For the first scenario, we need to introduce new policies that provide greater support to players born in the last quarter to mitigate Relative Age Effect (RAE). We need to further reflect on concrete procedures, such as learning for coaches, additional development stages for certain players, and other supportive measures. Indeed, Vaeyens et al.³⁴ explain that coaches often rely on their expertise and intuition to make subjective judgments, particularly when identifying and selecting talent. Acting only on them is not sufficient to reduce or eliminate Relative Age Effect (RAE) because coaches are often swayed by preconceptions and diverse pressures when choosing specific players.³⁵ Therefore, it is necessary to combine actions toward coaches, players, families, and structures to have an effect. Nevertheless, from a practical point of view, it is easier to reduce the maximum training capacity or increase the training of players who need it, rather than reducing the sensitivity. Therefore, $S_{11}$ or $S_{12}$ would be easier to implement than $S_{13}$ . In particular, $S_{12}$ would be preferred to $S_{13}$ since they both have the same reduction on the Relative Age Effect (RAE), but $S_{12}$ would be easier than $S_{13}$ . These findings align with recommendations from experts in talent identification and development,³⁶ which include delaying selection while providing targeted support for late developers, prioritizing the enhancement of both physical and mental skills over immediate competitive outcomes, and educating coaches as well as other key stakeholders.

In the second scenario, the first part, e.g. define categories based on a 15-month or 21-month period with a fixed starting month, is not consistent with our model because it modifies the length of the categories and, consequently, their number. The second part, e.g. rotating cut-off, does not change the number of cohorts, only the distribution of birth quarters within them, and each player remains in a category for the same duration as in the current system. Thus, this solution does not modify our modelization, and we can assume that the parameters remain applicable since the categories retain almost the same structure, except that the players who have more advantages change regularly. In scenarios, $S_{21}$ to $S_{24}$ , for year 1, the cut-off of a category is in January; in the second year, it is in April (July or October); in the third year, it is in January; and in the fourth year, it is in April (July or October), etc $\dots$ To implement this version, we simply need to assign a rate $α_{q}$ to each inflow and, if necessary, permute the inflow of experience for the first cohort. If the cut-off is in quarter $Q_{i}$ for $i \in [1 . . 4]$ , then for one year $α_{q} = \frac{1}{duration of cohort q}$ for $q \leq i$ and $1$ otherwise. For the second year, the values are inverted. For scenario $S_{25}$ , we alternate each case every year, ensuring a balanced distribution of advantages and disadvantages over time. For year 1, the cut-off of a category is in January; in the second year, it is in April; in the third year, it is in July; and in the fourth year, it is in October.

The cut-off date in April shows the worst improvement in Relative Age Effect (RAE), which is consistent with the fact that, in this case, we offer advantages to the first and second quarters, while the third and fourth quarters face disadvantages. The three other scenarios provide similar advantages, as they distribute the benefits more evenly across the birth quarters. However, relying solely on this indicator to classify the scenarios could necessarily lead to misinterpretations since the improvement in the Relative Age Effect (RAE) is assessed globally across all quarters, it does not necessarily imply an improvement in each individual quarter. For example, we might observe a significant improvement for the first quarter but little to no change for the fourth, leading to a high overall improvement despite disparities between quarters. Therefore, we need to analyze the proportion of players in each quarter for each scenario to properly assess its effectiveness and determine whether it leads to a more balanced distribution. Figures 7 and 8 illustrate this distribution for Scenario 1 and Scenario 2, respectively. In these figures, the dotted lines represent the birth quarters from INSEE, the gray lines represent the base case, and the colored lines represent the modified model. In each policy, the curve of quarters differs from the base case and tends toward the right proportion. For the first scenario, the effect applies in each quarter at the same time, and we observe the same efficiency as before for each scenario. $S_{14}$ and $S_{15}$ have almost the same distribution of birth quarters, even though quarter $Q_{4}$ is less favorable in the last policy. Figure 9 confirms this result. The area in this figure represents the difference between the INSEE and the model’s proportion of quarter births. We observe a significant reduction of this difference in scenarios 4 and 5 compared with the others. For scenario 2, the effect of the policies is more homogeneous since the advantage alternates between quarters. The Relative Age Effect (RAE) is nonetheless reduced and more so than in the first scenario for families. More advantages we give to the last quarter, more the curve aligns with the INSEE proportion, even if sometimes, as for quarter $Q_{2}$ in $S_{24}$ , the proportion becomes smaller than the objective. In the last scenario, we have higher efficacy. However, it should be noted that in this case, the Relative Age Effect (RAE) decreases until almost 2050 (see Figure 9) and then increases to reach an equilibrium. This is normal because when we alternate the cut-off date between all quarters, we give the same advantages to all quarters, and the model tends to give a proportion of 0.25, which is not the proportion given by INSEE. Thus, certain proportions tend toward their real values, and after a certain time, they exceed them.

Figure 7.

For each scenario from the $S_{1}$ family, the percentage of players for the four birth quarters is represented in color. The gray lines indicate the base case, while the dotted lines show the theoretical distribution of birth quarters based on INSEE data.

Figure 8.

For each scenario from the $S_{2}$ family, the percentage of players for the four birth quarters is represented in color. The gray lines indicate the base case, while the dotted lines show the theoretical distribution of birth quarters based on INSEE data.

Figure 9.

For each scenario, we plot the temporal evolution of the absolute values of the difference between the INSEE-reported birth quarter and the model-predicted one. Ideally, in the absence of Relative Age Effect (RAE), this difference would be zero.

To the best of our knowledge, there is no sport in which this method has been implemented. Nevertheless, other approaches relevant to our study can be found, such as bio-banding and birthday-banding. Bio-banding³⁷ consists of grouping athletes according to their biological age, determined by their maturity status. However, Cumming et al.³⁸ showed that this strategy moderates maturation bias but does not mitigate RAE.³⁸ A closely related method is birthday-banding,³⁹ which involves grouping athletes according to their date of birth rather than a fixed cut-off date: a player enters a cohort on their birthday and leaves it on their next birthday. This can be seen as our strategy applied at an individual level. Kelly et al.³⁹ examined this method within the England Squash talent pathway, where it has been in use, and observed no RAE in this sport, despite its presence in other racket sports (e.g., badminton,⁴⁰ table tennis,⁴¹ tennis⁴²). This study has some limitations, such as a small sample size. This method results in frequent player turnover within a cohort over the course of a year. While it may be feasible in individual sports such as squash, its implementation in team sports is far more challenging, as teams require stability and players need time to develop cohesion and coordinated play. Consequently, the continuous entry and exit of players based on their birthdays is incompatible with the practical demands of team dynamics. However, it supports our findings with the $S_{2}$ family.

Conclusion

In this article, we have built a dynamic model of the Relative Age Effect (RAE) that applies to French handball players. The model fits the data correctly and provides us with the opportunity to test different policies: some based on increasing attention to this phenomenon, and others based on modifying the cut-off date. The most efficient way is to modify the cut-off. This work provides an additional tool for decision-makers to consider when thinking about reducing the Relative Age Effect (RAE). This reduction can have an impact on talent detection and allow trainers to eventually have a wider sample of high-level players distributed across each quarter and it may also reduce attrition in young cohorts, contributing to a higher level of licensees. Indeed, we can assume that a portion of players who quit is due to the Relative Age Effect (RAE).^43,44 This result aligns with the findings of Pierson et al.⁹ in the National Hockey League (NHL) although these two models differ significantly in their construction.

Our model has some limitations. The parameters that have to be fitted are correlated, and the confidence interval becomes too wide. By construction, it will be difficult to express these parameters as independent parameters. Future work would be to analyze structural identifiability more precisely, despite the conceptual challenges posed by the mathematical and statistical complexity of the task. An overview of recent developments in the field of structural identifiability analysis can be found in Heinrich et al.⁴⁵ Nevertheless, we can certainly increase the precision if we use more values to fit them. Although we use a large sample of data to determine the constant parameters or the initial conditions, we only have 10 years of birth quarter proportions, from 2015 to 2024. The FFH does not have numerical records of license information before 2015. The initial conditions for the stocks of players were easy to extract from the data, but the initial conditions for the cumulative experience were not possible to estimate. We had to make some choices that seemed credible to us from a learning curve perspective and after discussions with coaches or trainers. The scenarios we proposed are not simple to implement in practice. For scenario 1, it will be difficult to change mentalities and habits since practices have always been focused on getting results quickly and therefore favoring the most advanced players. Nevertheless, we can assume that these scenarios will have the approval of parents who will see their children receive more attention and playing time. Moreover, it is difficult to estimate the relationship between changes in practice and the values of the model’s parameters. rc3The choice of values for the parameters $T S_{c}$ , $C o T_{c}$ , or $C o T_{c}^{m a x}$ is arbitrary, although they appear reasonable from a practical standpoint; however, a more in-depth analysis of their selection would be beneficial. In scenario 2, it is easier to show the impact of changes in the model. However, these changes will be difficult to implement from a practical organizational point of view for coaches, educators, and even parents. Although it is not an optimal model, it provides opportunities to anticipate policy changes. It will be improved in the future with the addition of supplementary data, always collected from the FFH. Another future work will be to complexify the model by introducing a more complex structure, such as the “pôle espoir” (youth development centers). These centers represent the majority of the flow from young to professional players. Even more complex, it would be interesting to create categories based on the maturity of the players, rather than their age, and observe the effect on Relative Age Effect (RAE). Finally, this model is not specific to handball, and it would be interesting to adapt it to other collective sports like football, rugby, etc. Only the initial values and fixed constants would need to be changed with the data from other sports.

Footnotes

Acknowledgments

The authors thank the reviewers for their valuable comments, which greatly improved the paper and the FFH for the provision of the data.

ORCID iDs

Cédric Noel

Jacques Prioux

Ethical considerations

Ethical approval for this study was waived as it involved the secondary analysis of pseudonymized administrative data. The dataset was formally provided by the French Handball Federation (FFHB) for research purposes. All data processing and storage were conducted in strict accordance with the General Data Protection Regulation (GDPR) and the French Data Protection Act (Loi Informatique et Libertés), specifically complying with the CNIL Reference Methodology MR-004 regarding the reuse of existing data for research. The researchers accessed only non-identifiable information (license identifiers and membership dates). No directly identifying data (names or contact details) were used, ensuring participant confidentiality. Under French regulations, formal IRB approval is not required for the retrospective analysis of such administrative datasets.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Data availability

The data analyzed in this study were provided by the French Handball Federation (FFHB) under a specific agreement and are not publicly available due to strict legal and ethical restrictions regarding participant confidentiality. The dataset remains the property of the FFHB. Any request to access these data should be directed to the French Handball Federation.

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From cut-off to kick-off: Dynamic modeling and mitigating of the relative age effect

Abstract

Keywords

Introduction

Material and methods

System dynamics

Data

Model structure

Coflow model

Physical training and reinforcing loop

Back to the model

Model calibration

Scenarios

Results

Parameters

Sensitivity analysis T S values

Testing scenario

Discussion

Model justification

Testing scenarios

Conclusion

Footnotes

Acknowledgments

ORCID iDs

Ethical considerations

Consent to participate

Consent for publication

Funding

Declaration of conflicting interests

Data availability

References

Sensitivity analysis $T S$ values