Atlas of the player pool

How much is a Superliga season worth in Premier League terms? The UEFA multiplier above answers that from countries' continental results; this model answers the same question from the players who actually changed leagues.

Movers — 2125 players observed in at least two leagues across 8108 qualifying player-seasons — anchor the model, since only a player's own before/after change of league separates their level from the league's scoring environment. A hierarchical Poisson model of non-penalty goals plus assists per 90 (Gelman et al., 2013) fits a league effect and a player effect together, sampled with NUTS (Hoffman and Gelman, 2014) in PyMC (Abril-Pla et al., 2023) and diagnosed with ArviZ (Kumar et al., 2019). Partial pooling keeps the league effects regularised while leaving player effects close to unpooled, the same within-subject logic behind plus-minus and RAPM ratings elsewhere in team sports (Kharrat, McHale and Peña, 2020; Hvattum, 2019).

In these terms, a Superliga season converts to 0.64 of a Premier League one (90 % HDI 0.58–0.70).

Refit on seasons before 2025/26, the model predicts each mover's first 2025/26 row after a league change — 865 such moves — against two baselines: the same rate as before, and that rate scaled by the ratio of UEFA multipliers. This is the same population CIES Football Observatory's expatriate-player reports track (Poli, Ravenel and Besson, 2024).

Against 18 leagues in common, the model's medians and the UEFA multipliers correlate at Spearman's rho = 0.72.

• HUN-NB I: model rank 6 vs UEFA rank 17 (m_L 0.74 vs multiplier 0.27).
• NED-Eredivisie: model rank 16 vs UEFA rank 8 (m_L 0.60 vs multiplier 0.51).
• NOR-Eliteserien: model rank 18 vs UEFA rank 13 (m_L 0.57 vs multiplier 0.37).

R-hat ≤ 1.005, minimum bulk ESS 1220, 0 divergent transitions across 8108 player-seasons from 2125 movers; fit in 223 s.

Posterior predictive check

Observed vs. replicated non-penalty goals plus assists per player-season (Vehtari, Gelman and Gabry, 2017): 15 % vs 14 % share of zeros, mean 4.45 vs 4.45, 90th percentile 11.00 vs 10.86.

What the movers do not show: they are not a random sample — players tend to move up when they are good and down when they are older — so the within-player contrast describes the league difference for the kind of player who moves, and the age term absorbs only the part of that selection that is age. The interval is the model's uncertainty, not the selection's.

The report's rankings keep the UEFA-coefficient multiplier throughout; the model above is shown alongside it as a check on that multiplier's own assumption — that continental results track player-level strength — not as a replacement for it.

What does a player's season tell us about the next one, given where he plays, how old he is and — for exports — at what age he moved? In model terms: predict a player's league-adjusted production next season (non-penalty goals plus assists per 90, quality-adjusted) from this season's numbers, for every player-season pair with at least 450 minutes in both seasons, every nationality in the corpus.

Evaluated by rolling-origin cross-validation (Hyndman and Athanasopoulos, 2021): for each of 5 target seasons in turn, every model trains only on pairs whose target season came earlier and is tested on that season's pairs — the same discipline a forecaster uses when the future is genuinely unknown at fit time, and what makes the per-season table below a real "performance over time" read rather than one blended number. Two baselines — persistence (next season = this season) and shrinkage to the league mean — sit alongside three fitted models sharing the same eight features: a hierarchical Bayesian regression (target ~ Normal(μ, σ), partial pooling on league and player, NUTS, 2 chains × 400 draws per origin (Gelman et al., 2013)), a gradient-boosted regressor and a small multilayer perceptron (scikit-learn (Pedregosa et al., 2011), hidden layers 64/32 — a small MLP, not an embedding model: PyTorch is not part of this pipeline). The Bayesian model's training rows are subsampled to at most 2500 per origin to keep five origins' worth of fits inside the runtime budget; the other four models train on the full split. The target is adjusted with the multiplier of the league the player is in next season; the features only know this season's league, so a move is a change the models cannot see coming — a limitation shared by all five.

Each cell: RMSE (MAE), in league-adjusted npG+A per 90. The first origin's training set is empty by construction — the corpus's earliest feature season leaves no earlier target season to train on — so only the two baselines are reported there.

The Bayesian model's 90 % predictive interval covered the observed value 92 % of the time, pooled across the 4 origins it was fit for (2022/23: 91 %, 2023/24: 90 %, 2024/25: 91 %, 2025/26: 93 %).

By pooled RMSE, Gradient boosting wins (0.071 vs 0.075 for persistence, 6 % lower).

The clearest season-to-season move in the winner's own RMSE is between 2023/24 and 2024/25 (0.005) — the kind of drift this rolling-origin table exists to surface.

The series is modelled as a local level in state space (Durbin and Koopman, 2012): the log of the season count follows a Gaussian random walk (σ ~ HalfNormal(0.2)), plus two ordered step changes δ₁, δ₂ in the level at unknown seasons τ₁ < τ₂ — on the series back to 1990/91 one step is misspecified, since the count rises through the 1990s and falls after the 2000s plateau, and a single step lands on whichever change buys more likelihood. NUTS only samples continuous parameters, so the τ pair is not sampled directly — every ordered pair of candidate seasons at least 3 seasons from either end and from each other is marginalised out of the model with a single log-sum-exp potential, and each break's own posterior is recovered afterwards from the continuous draws, the standard move for a marginalised discrete parameter (Gelman et al., 2013); the changepoint idea itself is due to (Adams and MacKay, 2007), applied here to a batch, two-break setting. The report's "break" is the step that lowers the level; the other is the rise. A rolling-origin backtest (Hyndman and Athanasopoulos, 2021) refits the same local level without the step at each of 15 origins, forecasting one season ahead and scoring against the naive "same as last season" baseline — the honest forecaster's read, since no real origin knows in advance which side of a break it sits on. The one forecast below, from the same change-point-free model fitted on the full series, is a demonstration of the method on a count of players, not a statement about any player.

For Denmark, the model dates the break to 2021/22 (23 % posterior probability), a ×1.41 (0.88–1.97, 90 % HDI) change in the level; the random walk's own innovation scale is σ = 0.081.

• 2021/22: 23 %
• 2020/21: 12 %
• 2022/23: 11 %

The same model, fit separately for the two contrast countries:

• Norway: 2004/05 (27 % posterior), ×0.66 (0.29–1.34). Rise: 2021/22 (23 %), ×1.32.
• Czechia: 2014/15 (56 % posterior), ×0.60 (0.39–1.03). Rise: 2000/01 (55 %), ×1.65.

One-step-ahead rolling-origin backtest of the plain local level (no change point) against the naive "same as last season", 15 origins from 2010/11 to 2024/25:

Pooled across 15 origins: MAE 4.60 for the model against 2.73 for the naive baseline, 100 % of the 90 % intervals covered the observed value.

The one forecast, for next season (2026/27), from the same change-point-free model fitted on the full series:

A demonstration of the method on a count of players, not a statement about any player.

Change-point fit: R-hat ≤ 1.004, minimum bulk ESS 372, 3 divergent transitions across 31 seasons.

For each of the 8 peer countries, the 2024/25 → 2025/26 average U21 share of domestic-league minutes (x) against the 2024/25 → 2025/26 average top-9 players per million (y) — one row per country. A Bayesian simple regression, y ~ Normal(α + β·x, σ), weakly informative priors (4 chains × 1000 draws (Abril-Pla et al., 2023)), answers the between-country question slide 3 asks: does a country with a higher average U21 share also have a deeper pool, on average (Gelman et al., 2013). Cross-checked against a plain pooled least-squares slope on the full two-season panel (numpy polyfit, ignoring country structure entirely, 1000 percentile-bootstrap resamples — statsmodels is not a dependency here), which should agree in sign. Two seasons per country (previous and metrics; the current season is partial and left out); a peer whose top flight FBref does not track is absent from the panel, which is why n can be below the peer count.

β = +10.11 per 10 percentage points of U21 share (90 % HDI −1.69–23.26), R² = 0.27; OLS on the same 8 country means lands close by, at +10.65 (−3.85–20.33), and the pooled-panel OLS slope agrees in sign at +7.76 (1.12–13.99). n = 8; the interval is wide because the panel is small.

R-hat ≤ 1.001, minimum bulk ESS 2126, 0 divergent transitions.

A separate check: the same regression, but with a country random intercept fit on the full two-season panel instead of country means. Within-country changes across the two seasons carry no signal — β_within = −0.16 per 10 percentage points (90 % HDI −2.53–2.17), n = 16 country-seasons (R-hat ≤ 1.004, minimum bulk ESS 851, 0 divergent transitions; posterior-median country-intercept scale σ_country = 5.19). With only two seasons per country the intercept absorbs almost all of the between-country pattern the fit above isolates, leaving this estimate driven by season-to-season noise alone; reported here as a robustness check, not a second headline.

Descriptive only: a country's average U21 share and its average per-capita count are shown together because they are the numbers on hand, not because one is claimed to produce the other.

Ridge regression (α = 1.00, standardised channels, converted back to original units) of top-9 players per million on three channels — U21 share of domestic minutes, domestic league strength (M2's m_L where the country's top flight is in that model's fitted set, else the UEFA-coefficient multiplier (every country in this run used the model estimate)) and median export age (recent entrants, or the all-time median for a country with none recently) — over the 8 peer countries with data on all three channels, metrics season. For one contrast country at a time, the fitted model's own prediction splits the gap exactly into one term per channel — the linear-model form of the Blinder-Oaxaca wage decomposition (Oaxaca, 1973; Blinder, 1973), applied here to a per-capita player count. Three linear channels means this split is Shapley-equivalent: the Shapley value of an additive term in a linear model equals its own coefficient's contribution, so no permutation machinery is implemented. The residual is whatever the three channels do not carry.

Bootstrap 90 % intervals on each channel's contribution, 1000 resamples of the panel's rows (the model refit on each resample; the home/contrast countries' own values held fixed); the residual itself is not bootstrapped — it is the two countries' own observed counts minus the fitted gap.

A decomposition of a correlation this panel happens to show, not a causal accounting; with n = 8 and three correlated national-level channels, the shares are indicative, not precise. A channel's share of the gap is only shown when the gap itself is at least 3 players per million — below that, a small denominator can send a share past 100 % in either direction; the contribution itself, in players per million, is always reported.

For every peer-nationality player (home nation included) whose first top-9-league season lies inside the fetched window and isn't censored (the same censoring rule as "Where do Danish players go when they leave?", reused here) — 115 players, ages 17–31 — age at that season is modelled against league-adjusted production over the player's first one or two top-9 seasons. f(age) is a natural cubic spline with knots at 19, 21, 23 and 25 (a plain quadratic below n = 150; the quadratic branch was used here), alongside origin-league strength (the transfer-graph model, § League strength), position and a partially pooled country effect (Gelman et al., 2013), sampled with NUTS (Hoffman and Gelman, 2014), 4 chains × 1000 draws; every design column and the outcome were standardised before fitting, the youth-minutes panel's own lesson about raw-scale priors on differently-scaled covariates.

β, per one-unit increase in origin-league strength (m_L): −0.03 (−0.12–0.05).

Danish exports' own country effect: +0.01 (−0.01–0.03); Danish exports arrive at a median age of 22.

The same model, fit again without origin-league strength: the country-effect scale (σ_n) is 0.138 with the league term in the model and 0.139 without it — what moves between the two is what the league term is absorbing.

Leave-one-nation-out: excluding Denmark's own 25 exports (n = 90 remaining) and refitting, the 21-vs-24 difference is 0.00 (−0.01–0.01), against 0.00 in the full fit.

R-hat ≤ 1.003, minimum bulk ESS 1128, 0 divergent transitions across 115 players; fit in 7.1 s.

What this model does not separate: the corpus is not a random sample of players who could have left later. Players who leave earlier tend to be the ones judged ready earliest — a selection effect the age curve mixes with any genuine development effect of arriving young, and this report does not try to tell the two apart.