Survey (deep-research, Jun 2026) anchoring the thesis: the skewness of the distribution of implied 1X2 returns is a per-league structural invariant, a function of competitiveness. Focus: modelling competitive balance. Each source has venue + why it matters. Claims verified adversarially (an N-0/N-1 vote indicates robustness).
The literature strongly supports half of the thesis — the mechanism
competitiveness → distribution of per-match p → skewness. But it contradicts
the strong form ("constant fixed in time + fluctuation = sampling noise"): the
two best EPL studies (Lee & Fort 2012; Basini et al. 2023) find real
structural breaks tied to institutional shocks (Champions League 94/95,
Bosman 95, revenue inequality).
Recommendation: reposition from "fixed constant" to "league-specific structural baseline, stable WITHIN competitive regimes". That is the form the evidence carries; the strong form it does not. Our finding F (the only "break" in 20 years = the dataset growing from 21→37 leagues in 2012) is already compatible with this — it just needs to be framed as intra-regime invariance, not absolute timelessness.
CB measurement is canonically a inequality/concentration problem, and each index has comparability flaws across leagues of different sizes — critical for us, since we compare leagues with different numbers of teams.
| Source | Venue | Why it matters |
|---|---|---|
| Utt & Fort (2002) | J. Sports Economics 3(4):367-373 | |
| Owen, Ryan & Weatherston (2007) | Review of Industrial Organization 31:289-302 | Bounds of raw HHI depend on league size; they propose normalised HHI (HHI*/dHHI) for cross-league comparability. → a defensible index for our panel. [3-0] |
| Borooah & Mangan (2012) | Applied Economics 44(9):1093-1102 | Generalised Entropy family (sensitivity parameter that re-weights parts of the performance distribution). → connects CB to moments/asymmetry of the distribution, a natural bridge to skewness. [3-0] |
For our W2: benchmark skewness against a size-robust index — HHI*/dHHI or SD-of-win-pct (Noll-Scully). Raw Gini is out (despite our current corr −0.83 using it informally — worth re-running with Noll-Scully/HHI*).
Here is the gold: formal machinery already exists that writes the per-match
(home/draw/away) distribution as a function of force/competitiveness — not only
the final season ranking. It is the template for deriving skewness = f(competitiveness).
| Source | Venue | Why it matters |
|---|---|---|
| Csató & Petróczy (2024) | arXiv:2406.19222 | Closest analogue to our thesis: ex-ante CB = (normalised) average win probability of the strongest team, via Elo W_ij = 1/(1+10^{-(R_i-R_j)/400}). It is competitiveness expressed as a function over P(strong wins). |
| Basini, Tsouli, Ntzoufras & Friel (2023) | JRSS-A 186(3):530-556 | Stochastic block model: the 1X2 result follows a multinomial whose parameters vary by force block (K×K×3 W/D/L array). → ties competitiveness-in-tiers directly to the 1X2 probabilities. Finds the EPL balanced until ~2003, imbalanced thereafter. [3-0] |
| Goddard & Asimakopoulos (2004) | J. Forecasting | Ordered-probit maps a latent force variable via 2 cut-offs into away/draw/home. A direct template for the distribution of p from force covariates. [3-0] |
| Koning (2000) | The Statistician / JRSS-D 49:419-431 | Ordered-probit of Dutch results used explicitly to study CB change over time — a precedent for match-level (not standings-level) CB analysis. [3-0] |
| Source | Venue | Finding |
|---|---|---|
| Lee & Fort (2012) | Scottish J. Political Economy 59(3):266-282 | Structural breaks split the EPL's history into 4 regimes; a sharp drop in the "Modern Period" aligned with Champions League 94/95, revenue inequality and Bosman 95. Regime change, not noise. [3-0; tri-causal attribution 2-1] |
| Basini et al. (2023) | JRSS-A | EPL balanced until ~2003, "quite imbalanced since then". [3-0] |
| Csató & Petróczy (2024) | arXiv:2406.19222 | ✅ The only strong ally for "fluctuation can be no-trend": with better measures, no long-run trend in the UCL group stage CB (2003/04–2023/24), overturning earlier studies that saw a decline. But it argues measurement artefact > pure sampling noise, and it is a tournament, not a national league. [2-1] |
Implication: our study must distinguish a FIXED-per-league skewness baseline from real regime breaks. That is exactly what blocks A (stationarity) and F (forensics of the 2012 break) attack — but the framing must acknowledge Lee & Fort and Basini head-on.
The favourite-longshot bias is the channel that converts the per-match distribution of p into skewness in the betting returns.
| Source | Venue | Why it matters |
|---|---|---|
| Whelan (2024) | Economica 91(361):188-209 | FLB present in fixed-odds football markets (not only pari-mutuel), generated by bettor disagreement + bookmaker risk aversion. → confirms the FLB in our own market. [3-0] |
| Golec & Tamarkin (1998) | J. Political Economy 106:205-225 | FLB as a preference for skewness — links return skewness to the shape of the bettor's utility, observationally equivalent to risk-love. |
| Snowberg & Wolfers (2010) | J. Political Economy / NBER WP 15923 | Via compound bets, they find that probability misperception (Prospect Theory) drives the FLB, not risk-love. |
Football-specific to close (cited in the corroboration, not verified here): Cain, Law & Peel (2000); Direr (2013); Angelini & De Angelis (2019) — the latter found the FLB weaker in recent European data, i.e. temporal variation in the bias itself that could mask/contaminate our invariance test. Worth finding and citing.
The choice of de-vig method moves the measured skewness directly, and the most common method is biased against precisely the signal we study.
| Source | Venue | Why it matters |
|---|---|---|
| Shin (1993) | Economic Journal 103(420):1141-1153 | Canonical structural de-vig model: endogenous spread (protection against insiders), separates true p from the margin via parameter z. Our primary method. [3-0] |
| Clarke, Kovalchik & Ingram (2017) | Am. J. Sports Science 5(6):45-49 | |
| Štrumbelj (2014) | Int. J. Forecasting 30(4):934-943 | Shin probabilities > basic normalisation/regression, in fixed-odds and exchanges. |
| Nash (2018) | arXiv:1811.12516 (preprint) | Formalises: the consensus price P_C ~ triangular around the true frequency P_T; corollary: P_T is distributed for each P_C. → a formal defence against the tautology "odds ARE p by definition", but also a constraint: each odd is consistent with a distribution of true competitivenesses. [3-0] |
The verification found no paper that:
- treats the skewness (3rd moment) of the de-vigged 1X2 distribution as the league-level invariant (the literature stops at the mean — Csató — or the variance/Noll-Scully);
- formalises
skewness = f(competitiveness)explicitly; - tests cross-league constancy with rigorous 3rd-moment / variance-ratio inference, facing the circularity of de-vigging head-on.
→ That is where the study enters. The methodological bridge (variance-ratio + 3rd-moment bootstrap applied to odds-skewness) is itself a contribution — no one has tied that machinery to a 3rd-order moment of odds.
- Circularity / tautology (the most serious). If "competitiveness" comes from the same odds that yield the skewness, the corr −0.83 is partly tautological. Defensible design: measure competitiveness from a source independent of the odds (HHI*/Noll-Scully from final standings, or Elo of results not prices) and skewness from de-vigged odds — and show that the link survives. Immediate action: re-run W2 with odds-independent competitiveness.
- De-vig sensitivity. Multiplicative vs power vs Shin diverge exactly in the
longshot tail that drives the skewness → run all 3 and show the corr is not a
method artefact (we already have
DEVIG_METHODparameterised). - Regime change vs invariance. Acknowledge Lee & Fort / Basini; reposition the thesis as intra-regime invariance (see TL;DR).
- League selection bias + variation in the FLB itself (Angelini & De Angelis): an FLB weakening over time may be confounded with skewness (in)variance.
The survey did not validate public availability beyond football-data.co.uk. Candidates to check before relying on them:
- football-data.co.uk extended — secondary divisions (E1-E3, SP2, I2, D2, F2…) beyond the main ones already used; more league coverage for the cross-section.
engsoccerdata(R package) — historical results for computing HHI/Gini/Noll-Scully per league-season (odds-independent source of competitiveness → attacks the circularity).- clubelo.com (API) — per-club Elo for Csató-style ex-ante CB, independent of prices.
- Kaggle "European Soccer Database" — results + multi-bookmaker odds.
- Opening vs closing odds (oddsportal/archives) — enables the "opening→closing drift" front from CLAUDE.md (needs opening odds).
Primary peer-reviewed: Utt & Fort, Owen et al., Borooah & Mangan, Basini et al., Goddard & Asimakopoulos, Koning, Lee & Fort, Whelan, Golec & Tamarkin (via Snowberg & Wolfers), Snowberg & Wolfers, Shin, Štrumbelj. Preprints: Csató & Petróczy (peer-review-track), Nash (single-author, not reviewed). Low-tier but useful: Clarke et al.