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REVIEW 3 major objections 7 minor 73 references

Football is becoming more predictable; Network analysis of 88 thousands matches in 11 major leagues

T0 review · 3 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Top-division European football has become more predictable over the 26 seasons from 1993-94 to 2018-19.

desk verdict A useful panel measurement of football predictability, but the headline trend may be an artifact of the paper's decision to drop draws; worth refereeing, not taking at face value. read the letter →

arxiv 1908.08991 v2 pith:H3IN7PX2 submitted 2019-08-23 physics.soc-ph cs.SIstat.OT

Victor Martins Maimone , Taha Yasseri This is my paper · ORCID
classification physics.soc-phcs.SIstat.OT
keywords footballpredictabilityeigenvectorcentralitynetworkanalysiscompetitivebalancehome-fieldadvantageGinicoefficientsportsforecasting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that top-division European football has become more predictable over the 26 seasons from 1993-94 to 2018-19. It measures predictability with a deliberately self-contained model that uses only past results: matches are turned into a directed network with edges from loser to winner weighted by points, team strength is read from eigenvector centrality, and the home-minus-away score difference is fed into a logistic regression that is scored by area under the ROC curve. Across 87,816 matches and 11 major leagues, the paper finds rising predictability in most leagues, with all of them tending toward an AUC near 0.75. It reports two supporting trends—greater inequality in final points and a shrinking home-field advantage—and argues they are consistent with money concentrating success in the same clubs, while stating explicitly that the causal link to monetization is not directly tested.

What carries the argument

The central mechanism is a directed match network whose eigenvector centrality serves as a self-contained team-strength rating. In the network, each edge points from the loser to the winner and is weighted by the points the winner earned, and a team's eigenvector centrality reflects not only how often it wins but how strong its beaten opponents are—a team is central if it beats teams that are themselves central. The difference between the home and away teams' centrality scores is the single input to a logistic regression, and the area under that regression's ROC curve on matches outside the training window is the paper's operational measure of predictability. The model is intentionally simple and time-consistent: it uses only past results and fixed parameters, so any historical trend in its AUC reflects changes in the game itself rather than improvements in the predictor.

What would settle it

Re-run the same centrality-based model on all 87,816 matches using three outcomes—home win, draw, and away win—and compare the AUC trend with the paper's two-outcome trend; if the rise in predictability disappears once draws are included, the central claim is specific to decisive matches, not to football generally.

Watch

Extended reading notes

Core claim

The central claim is that football's results have become easier to anticipate, not just that some teams are stronger. The authors define predictability as the out-of-sample AUC of a fixed, past-results-only model, so an upward trend in AUC means the same simple predictor extracts more information from recent results now than it did in the 1990s. They find that seven of the eleven leagues studied show increasing AUC over the sample, Greece and Turkey show the opposite but with recent increases, Belgium and Italy are stable, and the leagues tend to converge near 0.75 AUC. The paper also shows that the Gini coefficient of end-of-season points is positively correlated with AUC in every league, and that home-field advantage, measured both by the model's logistic offset and by the historical share of home points, has declined in all eleven leagues. It explicitly does not claim to have established the monetary cause of these trends.

Load-bearing premise

The load-bearing assumption is that removing drawn matches does not distort the historical comparison; if draws changed in frequency or predictability over 26 years, the upward trend measured on matches with a winner may not describe football as a whole.

Editorial extensions

If this is right

  • Because the prediction model uses only past results and its parameters are fixed, the rising AUC cannot be attributed to smarter forecasting; the same simple model has become a better predictor of future results over time.
  • The positive correlation between AUC and Gini coefficient in all 11 leagues indicates that inequality in final points and predictability move together, so leagues with more concentrated results are also the leagues where outcomes are easier to forecast.
  • The universal decline in home-field advantage means that home status contributes less to match outcomes than it used to, which reduces the number of matches a weaker team can win simply by playing at home.
  • The observed convergence of league AUC values near 0.75 means that differences between leagues in how predictable their results are have shrunk even while the overall level of predictability has risen.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the tie-removal assumption would be to include draws as a third outcome; if the AUC trend disappears, the paper's result holds only for matches with a winner, not for football as a whole.
  • The paper's gentrification feedback loop can be tested with club finances: within a league, seasons with larger gaps in wage bills or transfer spending between top and bottom clubs should show higher AUC than seasons with smaller gaps.
  • Because the methodology is sport-agnostic, applying the same network-AUC measure to leagues with salary caps would disentangle whether the predictability trend comes from unrestricted spending or from something intrinsic to football; the paper names this comparison as future work.
  • The model's own home-advantage parameter offers a way to separate the two trends the paper reports: a regression of league-season AUC on league-season home advantage would show whether the shrinking home boost is actually one of the mechanisms making results easier to predict.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. This paper uses 87,816 matches from eleven top European leagues between the 1993/94 and 2018/19 seasons to ask whether football outcomes have become more predictable. The authors build two self-contained models—a simple points-share difference and a network eigenvector-centrality score computed from past matches—and benchmark them against Bet365 odds. After excluding drawn matches, they fit a logistic regression of home-win/away-win on the score difference and measure predictive performance with the Brier score and AUC. They report a general upward trend in per-season AUC (with Greece and Turkey as exceptions), convergence toward AUC around 0.75, a positive correlation between AUC and the Gini coefficient of season points, and a decreasing home-field advantage across all leagues, which they interpret as consistent with a 'gentrification' of football.

Significance. The study is a valuable first large-scale, historically consistent attempt to quantify predictability in football with a simple network-based model, and the public data source and fixed model specification make the approach transparent and reproducible in principle. The decline in home-field advantage is a robust and interesting side result. However, the central trend claim currently rests on an unconditioned two-class analysis of non-draw matches and on visually assessed time trends without inferential support; these issues must be resolved before the main conclusion can be accepted.

major comments (3)
  1. [Modelling predictability] The decisive modelling choice is to 'limit our analysis to the matches that have a winner and eliminate the ties from the entirety of this study.' Because draws are a large, non-random fraction of football matches, removing them can change the apparent difficulty of the classification problem even if the full three-outcome distribution is unchanged: if the draw rate fell over time, the retained subset would contain progressively more lopsided matches, and AUC on that subset could rise mechanically. The paper never reports draw rates by league and season, never estimates a three-outcome (home/draw/away) model, and never checks whether the reported AUC trend survives adjustment for draw frequency. The authors should add these analyses or explicitly demonstrate that the non-draw AUC trend is not an artefact of selection.
  2. [Results and Discussion: Predictability over Time] The central 'becoming more predictable' claim is supported only by lowess curves and visual inspection in Figure 4. AUC for a single league-season is estimated from a few hundred matches and will carry substantial sampling noise, yet no confidence intervals, trend coefficients, significance tests, or adjustments for temporal autocorrelation are reported. The authors should supply formal trend estimates for each league (for example, regression or rank-based trend tests with season as the covariate) and, given the autocorrelated series, a time-series-aware inference procedure. The statement that 'all leagues tend to converge towards 0.75 AUC' likewise needs quantitative support.
  3. [Model Performance] The text says the models and the betting-market benchmark are 'statistically indistinguishable at the 2% significance level for the majority of year-leagues,' but Tables S1 and S2 contradict this. In Table S2, eight of eleven AUC comparisons between the Network model and the Market are significant at p<0.02 (England, Germany, Spain, Italy, Portugal, Netherlands, Belgium, France), all favouring the market, and several Brier-score comparisons are also significant at the 2% level. This is an internal inconsistency in a passage used to justify the model as a reliable measurement tool. The comparison should be re-run or, at minimum, described accurately in the text.
minor comments (7)
  1. [Modelling predictability, Eq. (1)] Logistic-regression parameters are not obtained by 'ordinary least squares methods'; the standard estimation is maximum likelihood via iteratively reweighted least squares. Please correct the wording.
  2. [Fig. 1 caption] The caption says the network is shown 'after 240 matches have been played for n = 0.5' but also says the centrality is calculated from 'the last 190 matches'; for a 380-match season, n = 0.5 gives 190 matches, so one of these numbers is wrong.
  3. [Fig. 4] Please state the smoothing span used for the lowess fits and clarify whether the plotted points are raw per-season values or smoothed values; the dual-axis display currently makes it hard to assess the data behind the curves.
  4. [Table 1] The AUC–Gini correlations need a stated method (Pearson or Spearman; raw or smoothed series) and confidence intervals or p-values; the value 0.413 for Belgium should be described as weak-to-moderate, not 'high.'
  5. [Eq. (2)] Because draws are excluded, the market probabilities in Eq. (2) are conditional on no draw; please state this explicitly, since the raw Bet365 odds include a third outcome and the normalization is not the usual three-outcome calibration.
  6. [Conclusion] The limitations paragraph acknowledges data volume and model sophistication but omits the draw-exclusion choice, which is the most consequential modelling decision for the main claim; this should be discussed and, ideally, checked.
  7. [Throughout] There are numerous typos and OCR-like artifacts (for example, 'predictibility' in the Results section, '88 thousands' in the title/abstract, 'Ho e Points' in Figure S8, 'T urkey' in Table 1) that a careful proofreading pass should remove.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central AUC measure is self-contained, externally benchmarked, and not equivalent to its inputs by construction.

full rationale

No significant circularity found. The central measure, AUC of a binary home-win/away-win classifier, is computed from features (score differences built from past matches) and outcomes (the result of the match being evaluated), so the target variable is not an input to the feature construction. The logistic regression in Eq. 1 is a standard fit; the network model uses eigenvector centrality from previous results; and the betting-market benchmark in Eq. 2 is external to the authors' models. The Gini coefficient is reported as a correlate of the AUC trend, not as a fitted input that defines or forces the AUC. The paper's decision to exclude draws is a validity limitation rather than a circular step: the AUC trend is defined on the non-draw subset, so the conclusion is conditional on that subset, but the measure is not equivalent to its inputs by construction. The paper's own limitation statements in the Conclusion, concerning sample size and deliberately simple self-contained models, are acknowledged scope restrictions rather than admissions of circular reasoning. Self-citations are not load-bearing; the cited centrality and Gini methods are standard external results, and the market benchmark provides an independent reference point. The reported trend, even if debatable on selection-bias grounds, follows from an empirical calculation rather than from a definitional tautology or a fitted-parameter-as-prediction step.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central analysis rests on an external match database, a binary framing that drops draws, a network centrality definition of team strength, and a market-odds benchmark. None of these is independently validated in the paper, but they are standard modeling choices rather than invented entities. The only hand-chosen quantity is the training window n=0.5, with sensitivity checks in the SI.

free parameters (2)
  • Training window fraction n = 0.5 (main analysis); robustness at 0.1, 0.3, 0.7, 0.9 in SI
    The model uses the past n=N/T fraction of a season to estimate team strength. The authors select n=0.5 for all reported trends; accuracy magnitude changes with n, though the SI shows the comparison.
  • Logistic regression parameters (mu, s) = Per league, season, and model
    Equation 1 parameters are estimated from score differences and outcomes. The AUC curves and resulting trends depend on these fits. The text says they are obtained by ordinary least squares, which is not the standard logistic-regression estimator.
assumptions (5)
  • domain assumption The football-data.co.uk database is accurate, time-consistent, and free of systematic bias across 11 leagues and 26 seasons.
    All calculations depend on this external source; the paper does not cross-validate against another data provider.
  • domain assumption Drawn matches can be removed without biasing the measurement of predictability.
    The paper states 'we eliminate the ties from the entirety of this study' without testing whether draw rates or draw predictability changed over time.
  • domain assumption Eigenvector centrality on the loser-to-winner directed network is a valid proxy for team strength.
    The network measure is assumed to handle strength-of-schedule better than raw points; support is only the benchmark comparison, which is partially inconsistent.
  • domain assumption Normalized Bet365 odds provide a good estimate of state-of-the-art market probability.
    Equation 2 normalizes home and away odds while dropping draw odds; the paper treats this as a strong benchmark for model validation.
  • standard math A logistic function adequately links team-strength difference to win probability.
    Equation 1 assumes a sigmoidal relationship; this is standard and not central to the trend claim.

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Pith. "Pith review of Football is becoming more predictable; Network analysis of 88 thousands matches in 11 major leagues." pith.science (2026). https://pith.science/paper/H3IN7PX2

@misc{pith2026190808991,
  author       = {Pith},
  title        = {Pith review of: Football is becoming more predictable; Network analysis of 88 thousands matches in 11 major leagues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H3IN7PX2}},
  note         = {Machine review of arXiv:1908.08991}
}
read the original abstract

In recent years excessive monetization of football and professionalism among the players has been argued to have affected the quality of the match in different ways. On the one hand, playing football has become a high-income profession and the players are highly motivated; on the other hand, stronger teams have higher incomes and therefore afford better players leading to an even stronger appearance in tournaments that can make the game more imbalanced and hence predictable. To quantify and document this observation, in this work we take a minimalist network science approach to measure the predictability of football over 26 years in major European leagues. We show that over time, the games in major leagues have indeed become more predictable. We provide further support for this observation by showing that inequality between teams has increased and the home-field advantage has been vanishing ubiquitously. We do not include any direct analysis on the effects of monetization on football's predictability or therefore, lack of excitement, however, we propose several hypotheses which could be tested in future analyses.

Figures

Figures reproduced from arXiv: 1908.08991 by the authors.

Figure 1
Figure 1. The network diagram of the 2018-2019 English Premier League after 240 matches have been played for n = 0.5 (calculating centrality scores based on the last 190 matches). Network Model. To overcome the above mentioned limitation and come up with a scoring system that is less sensitive to the set of teams that each team has played against, we build a directed network of all the matches within the training window, in w… view at source ↗
Figure 2
Figure 2. Logistic regression model example for the England Premier League 2018-2019 for the two models and the benchmark model. "'#%/( )#' )! , ) " ",( )2 ,"" " . '2 ".$",' )!- *,./# ' *.' )! + %) /,&"2 */).,2         0", #",%", *," 0", #",%", *,"  .* ) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Accuracy of the models measured through a)average Brier Score and b) area Under the Curve (AUC), per league. Model Performance. To compare our two models with the benchmark model, we calculate a loss function in the form of the Brier score for all the (1 − n)T matches in different leagues and years from 2005 to 2018 and n = 0.5 (See Data and Methods for details). The distributions of scores are reported in Figure S1… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Time Trends In AUC and Gini Coefficient. Blue dots/lines depict AUC and are marked in the primary (left) y-axis; Orange dots/lines depict the Gini coefficient and are marked in the secondary (right) y-axis; Both lines are fitted through a lowess model. 4 [PITH_FULL_IM…
Figure 5
Figure 5. Figure 5: The Decrease in Home Field Advantage. Left: Home field advantage calculated by the two models. Right: The share of home points measured on historical data. The straight lines are linear fits. See the detailed graphs for each league in Figures S8–S11. Increasing Inequal…

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00AUC AUC; Greece

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.60 0.65 0.70 0.75 0.80 0.85 0.90AUC AUC; Italy

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95AUC AUC; Netherlands

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.60 0.65 0.70 0.75 0.80 0.85 0.90AUC AUC; Portugal

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.5 0.6 0.7 0.8 0.9AUC AUC; Scotland

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90AUC AUC; Spain

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    Network 3

    Dyadic 2. Network 3. Market Prediction Model 0.5 0.6 0.7 0.8 0.9AUC AUC; Turkey Fig. S2. Area Under the Curve (AUC) distributions per country for the benchmark and prediction models (for n = 0 .5). 10 BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainT...

    2005

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    Market BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90Average AUC Average AUC: 2005/06 to 2018/19 (n = 0 .1)

    2005

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    Market Fig. S3. Average Accuracy per Country: n = 0 .10 BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0 .10 0.12 0.14 0.16 0.18 0.20 0.22 0.24Average Brier Score Average Brier Score: 2005/06 to 2018/19 (n = 0.3)

    2005

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    Market BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90Average AUC Average AUC: 2005/06 to 2018/19 (n = 0 .3)

    2005

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    Market Fig. S4. Average Accuracy per Country: n = 0 .30 11 BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0 .10 0.12 0.14 0.16 0.18 0.20 0.22 0.24Average Brier Score Average Brier Score: 2005/06 to 2018/19 (n = 0.7)

    2005

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    Market BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90Average AUC Average AUC: 2005/06 to 2018/19 (n = 0 .7)

    2005

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    Market Fig. S5. Average Accuracy per Country: n = 0 .70 BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0 .10 0.12 0.14 0.16 0.18 0.20 0.22 0.24Average Brier Score Average Brier Score: 2005/06 to 2018/19 (n = 0.9)

    2005

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    Market BelgiumEnglandFrance GermanyGreece Italy Netherlands PortugalScotland SpainTurkey Country 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90Average AUC Average AUC: 2005/06 to 2018/19 (n = 0 .9)

    2005

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    Market Fig. S6. Average Accuracy per Country: n = 0 .90 12 19952000200520102015 Time 0.14 0.16 0.18 0.20 0.22Brier Score Brier Score; England Linear Fit Lowess Fit Scores 1995 2000 2005 2010 2015 Time 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24Brier Score Brier Score; Spain L...

    1995

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.