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Structural Stability of Skewness in Football Betting Markets

Status: consolidated methodology · pipeline reproduced on a 2005–2025 sample Author: Vitor Alves Dataset: frozen (data/PROVENANCE.json, sha256 6905ca53…), 205,435 matches, 38 leagues, 2005-01→2025-06. Target: Royal Society Open Science (empirical regularity + mechanism + reproducible artefact).

1. Research question

Is the skewness of the distribution of implied returns in football betting markets a temporal process or a structural invariant? Three fronts:

  1. Cross-sectional — where does the skewness come from? (mechanism)
  2. Temporal — does it have dynamics (drift, persistence, breaks) or is it constant?
  3. Structural — what determines the level of each league?

Thesis: skewness is, to first order, the algebraic image of the distribution of implied probabilities; the level of each league is determined by (slow) sporting competitiveness, which makes it a temporal invariant. The bookmakers' margin is orthogonal to the asymmetry.

2. Gap in the literature

Golec & Tamarkin (1998) treat the preference for skewness as a cross-sectional phenomenon (across bets, within a cross-section); the favourite-longshot bias (FLB) is the mechanism (Snowberg & Wolfers 2010). Missing are (i) the mechanical decomposition quantifying how much of the market skewness is the Bernoulli identity of the distribution of p; (ii) the structural law skewness=f(competitiveness) measured with a regressor independent of the odds; (iii) the temporal invariance over 20 years as a statement about efficiency/microstructure. This is the contribution.

3. Data

  • Source: normalised football-data.co.uk (xgabora mirror; main + extra/South American leagues), frozen by hash. Cut-off ≥2005 (odds coverage ~100%).
  • N: 205,435 1X2 matches; 148,261 with an over/under 2.5 market.
  • Columns: OddH/D/A (closing average) and MaxH/D/A (best price → margin test); FTResult, FTHome/FTAway (goals → O/U and Elo); HomeTeam/AwayTeam (Elo); Division, MatchDate.

4. Primary object — ex-ante (de-vigged) skewness

A unit bet on the favourite at decimal odds o with true probability p: return (o−1) with prob. p, −1 with prob. 1−p — a rescaled Bernoulli with closed-form central moments (μ=po−1, σ²=p(1−p)o², m₃=p(1−p)(1−2p)o³). The per-match skewness depends only on p: (1−2p)/√(p(1−p)), crossing zero at p=0.5. The aggregate skewness (league/window) is that of the mixture, decomposed by the law of total cumulants:

M₃ = E[m₃ᵢ]                     (mechanical: within-match asymmetry / FLB)
   + 3·E[σ²ᵢ(μᵢ−μ)]             (variance×mean covariance)
   + E[(μᵢ−μ)³]                 (between-match dispersion)
  • De-vig: Shin (1993) primary (by-product z = fraction of informed money); multiplicative and power as robustness.
  • Realised ex-post (skewness of the actual returns) = robustness; should converge to the ex-ante under calibration.

Multi-moment extension (shape). The rescaled Bernoulli has closed-form central moments of every order, m_k = oᵏ·p(1−p)·[(1−p)^{k−1} + (−1)ᵏ·p^{k−1}], and the k-th moment of the mixture follows from the law of total moments, M_k = E_i[Σ_j C(k,j)·m_{j,i}·dᵢ^{k−j}], dᵢ=μᵢ−μ (the decomposition of M₃ above is the k=3 case). This measures var/skew/kurtosis/5th–6th order of the implied distribution and the within fraction (mechanical) per order. Under fair odds the means are zero (d≡0), so M_k=E[m_k] and the ordered-probit predicts each league moment from competitiveness (not only the 3rd) — shape invariance. The distribution collapse (KS conditional on the p_fav band; the effect size is the KS statistic, since the p-value saturates with large n) tests whether, holding competitiveness fixed, the distribution is the same across leagues.

5. Odds-free competitiveness (breaking the circularity)

Measuring competitiveness via p_fav (from the odds) is circular. We build a results-only Elo: chronological multi-league step (W/D/L + goal difference, home advantage), and a rating-diff→(P_H,P_D,P_A) map via an MNLogit calibrated on the results. Per-league measures: mean forecast entropy, Elo favourite probability, force dispersion, upset rate. None touches odds.

6. Temporal panel

Unit = (league, season) — dissolves the composition confound by construction. Tests: secular trend (league FE + year, cluster SE); between/within decomposition with a sampling-noise benchmark (bootstrap of matches); per-league trends/breaks; COVID vignette (empty stadiums in 2020 as a natural experiment of a shock to home advantage).

7. Tests

Dimension Test
De-vig calibration over-rate vs p_over; ex-ante vs ex-post
Decomposition law of total cumulants (within/cov/between)
Odds-free mechanism corr/OLS skew~Elo + bootstrap CI (n=38)
Stationarity/i.i.d. ADF+KPSS, Ljung-Box, Variance-Ratio, AR(1)
Temporal invariance panel FE+year (cluster SE), ICC, breaks
Margin overround and skew: average vs maximum odds
Closed form (E1) quadrature of the Gaussian integral vs MC; near-balance expansion
Force robustness (E2) skew×p_fav curve under t-Student/skew-normal/uniform
Robustness de-vig (mult/power/shin), window, overlap, binary O/U

8. Results (frozen sample)

Finding Value Reading
Global ex-ante / ex-post skew +0.236 / +0.230 implied object reproduces the realised
M₃ decomposition +102.6% within-match, ~0% between-match skewness = algebraic image of the FLB
corr(elo_pfav, p_fav_odds) +0.909 odds read off sporting competitiveness
skew ~ odds-free competitiveness +0.83 (upset) / −0.75 (elo_pfav) non-circular law survives
Secular trend (panel) β=+0.00015/year (p=0.73) no drift over 20 years
ICC (between/total) 0.70 league invariant dominates over time
Margin: overround vs skew 1.067→1.009 vs +0.236→+0.254 margin orthogonal to asymmetry
Binary O/U 2.5 ex-ante −0.210 (within 99.6%) identity holds beyond 1X2
Closed form S(σ_L) (E1) quadrature ≈ MC (max|Δ|=0.0015); S₀=(1−2p₀)/√(p₀(1−p₀)) law is a closed integral, not a simulation
Force robustness (E2) max|ΔS|=0.03 (t/skew-normal/uniform) < league-sd 0.05 law = geometry of the mixture, not Gaussianity

Synthesis

The skewness of the betting market is a structural invariant: ~100% the within-match asymmetry of the distribution of probabilities (mechanical), with a league level determined by sporting competitiveness — a relationship that survives a competitiveness measure independent of the odds — and with no temporal drift over 20 years. The bookmakers' margin affects the level of return, not the asymmetry. The risk asymmetry is inherited from the sport, not produced by the pricing.

9. Core references

  • Golec & Tamarkin (1998). Bettors Love Skewness, Not Risk, at the Horse Track. JPE.
  • Snowberg & Wolfers (2010). Explaining the Favorite-Longshot Bias. JPE.
  • Shin (1993). Measuring the Incidence of Insider Trading in a Market for State-Contingent Claims. Economic Journal. (de-vigging)
  • Štrumbelj (2014). On determining probability forecasts from betting odds. IJF.
  • Andrikogiannopoulou & Papakonstantinou. Estimating Risk Preferences from Betting Choices.
  • Constantinou & Fenton (2012). Solving the problem of inadequate scoring rules for assessing probabilistic football forecasts. (Elo/probabilities)
  • Kraus & Litzenberger (1976); Harvey & Siddique (2000); Barberis & Huang (2008) — skewness in finance (contrast).