Vitor Alves — Cold Code Labs ORCID: 0009-0008-3522-1694 Correspondence: vitor@coldcodelabs.com
Target: Royal Society Open Science (working draft)
We ask whether the skewness of returns implied by football betting markets is a temporal process or a structural invariant. Using 205,435 matches across 38 leagues (2005–2025), we define the ex-ante skewness of the favourite bet from de-vigged (Shin) implied probabilities. A law-of-total-cumulants decomposition shows market skewness is ≈100% the within-match Bernoulli asymmetry of the win-probability distribution — the algebraic image of the favourite–longshot bias (ex-ante +0.236 ≈ ex-post +0.230; within-match term +102.6%). Its cross-league level tracks competitiveness and survives a competitiveness measure built without odds (results-only Elo; skewness vs upset rate r = +0.83), while odds- and Elo-based favourite strength agree at r = 0.91. A league×season panel shows no secular trend within the modern regime (β = +0.00015/yr; a single break, COVID-19; 70% of variance is between-league); by an equivalence test (two one-sided tests) the trend is statistically equivalent to zero within half a between-league SD (p = 0.006), turning the usual high p-value into positive evidence of stability. Margin is orthogonal to asymmetry, and the identity reappears in the binary over/under-2.5 market. Cast as one instrument for the similarity of asymmetries, a single competitiveness parameter explains 82% of the between-league variance, the first two moments 98%, and the full win-probability distribution is the minimal sufficient statistic. Finally, ported unchanged through a sport-agnostic data layer, the same law holds out of sample in tennis and basketball — two-outcome markets with independent odds sources (corr(skew, competitiveness) −0.95 to −1.00; underdog skew +2.3 to +2.6) — identifying it as a property of competition rather than of football or the 1X2 contract. The market's risk asymmetry is a structural baseline inherited from the sport, not produced by pricing it.
Keywords: skewness; favourite–longshot bias; betting markets; market efficiency; implied probabilities; structural invariance; equivalence testing; cross-sport validation.
Betting markets are a long-standing laboratory for studying how prices encode beliefs and preferences, because each contract has a known, exogenously determined terminal payoff [Thaler & Ziemba 1988; Snowberg, Wolfers & Zitzewitz 2013]. A central, well-documented regularity is the favourite–longshot bias (FLB): longshots are systematically over-priced and favourites under-priced relative to their empirical win frequencies [Snowberg & Wolfers 2010]. A leading explanation is a preference for skewness: bettors accept a lower expected return in exchange for the small-probability, large-payoff (positively skewed) profile of a longshot [Golec & Tamarkin 1998; Garrett, Hartley & Coughlin 2006; Andrikogiannopoulou & Papakonstantinou 2020]. This literature is overwhelmingly cross-sectional: it characterises how skewness varies across bets within a single cross-section.
We ask a different, complementary question. Treat the skewness of the market's return distribution as a quantity that can be measured repeatedly over time and across competitions. Is it a temporal process — drifting, mean-reverting, regime-switching — or a structural invariant fixed by the sport? Three sub- questions follow: (i) where does the skewness come from (mechanism); (ii) does its time series carry dynamics; and (iii) what determines its level in a given league?
Our contribution is threefold. First, we make explicit and quantify the mechanical core: the skewness of a fixed-odds bet is, by construction, an algebraic function of the win probability, and we show via a moment decomposition that essentially all of the aggregated market skewness is this within-match term — the FLB structure — rather than an artefact of pooling heterogeneous matches. This converts the natural objection ("isn't this tautological?") into the result itself. Second, we establish the structural law skewness = f(competitiveness) using a competitiveness measure constructed without any market information (a results-only Elo), breaking the circularity that would otherwise undermine the claim. Third, we document intra-regime temporal invariance as a statement about market efficiency and microstructure: despite the enormous growth of online betting, exchanges and algorithmic pricing over this period, the risk-asymmetry structure of the market did not move. We are careful here: the competitive-balance literature documents real regime breaks in English football tied to institutional shocks — the Champions League expansion (1994/95), the Bosman ruling (1995) and rising revenue inequality around 2003 [Lee & Fort 2012; Basini et al. 2023]. These all predate our ≥2005 sample, which therefore sits within a single modern regime; our claim is invariance of a league-specific baseline within that regime, not atemporality, and we test for within-window breaks directly.
Related work and the gap. Three strands meet here. The competitive-balance literature measures imbalance with size-robust indices — the Noll-Scully ratio, the normalised HHI* of Owen, Ryan & Weatherston (2007) and generalised-entropy families [Borooah & Mangan 2012] — and warns that the Gini coefficient is invalid for zero-sum league play [Utt & Fort 2002]; but it stops at the first and second moments of the outcome distribution. The forecasting literature provides the bridge from strength to per-match outcome probabilities — ordered-probit models [Goddard & Asimakopoulos 2004; Koning 2000] and strength-block multinomial models [Basini et al. 2023] — and the nearest analog to our object, Csató & Petróczy (2024), expresses ex-ante balance as the (normalised) mean win probability of the stronger side, finding, like us, no long-run trend, but for tournament group stages and at the mean rather than the third moment. The de-vigging literature supplies our tools and a caution: the multiplicative method ignores the favourite–longshot structure [Clarke, Kovalchik & Ingram 2017], Shin's method is strong but not universally dominant [Shin 1993; Štrumbelj 2014], and each price is formally consistent with a distribution of true probabilities rather than a single one [Nash 2018] — a defence against the charge that "odds are probabilities by definition." To our knowledge no prior work treats the skewness (third moment) of the de-vigged 1X2 distribution as a league-level invariant, formalises skewness = f(competitiveness), or tests its cross-league constancy with explicit third-moment inference while confronting de-vig circularity head-on. That is the gap we fill.
We use a normalised, multi-league compilation of football-data.co.uk match data
(2000–2025), restricted to seasons ≥ 2005, from which odds coverage is
essentially complete. The analysis sample is 205,435 matches across 38 leagues
(top and second divisions across Europe, plus South-American, North-American and
Asian competitions). For each match we observe the 1X2 closing odds (market
average, OddH/D/A, and best market price, MaxH/D/A), full-time result and
goals, team identities, division and date. A subset of 148,261 matches also
carries over/under-2.5-goals odds. The dataset is frozen by content hash
(PROVENANCE.json) and the full pipeline is reproducible from a single command.
Invalid odds (≤ 1.01) are dropped.
A unit stake on the favourite at decimal odds o, with true win probability p,
returns o − 1 with probability p and −1 with probability 1 − p. This is a
re-scaled Bernoulli variable, with closed-form central moments
- mean
μ = p·o − 1, - variance
σ² = p(1 − p)·o², - third moment
m₃ = p(1 − p)(1 − 2p)·o³,
so that the per-match skewness depends only on p:
skew(p) = (1 − 2p) / √(p(1 − p)),
crossing zero at p = ½, positive for longshots (p < ½) and negative for strong favourites (p > ½). The odds o set the magnitude of the winning payoff but, by affine invariance, do not affect the skewness. This identity is the analytic backbone of the FLB.
Bookmaker odds embed a margin (overround): the raw inverse-odds sum to more than one. To recover probabilities we de-vig using the Shin (1993) model, whose single parameter z estimates the proportion of informed ("insider") money [Štrumbelj 2014]; we use multiplicative (proportional) and power de-vigging as robustness, and report sensitivity across all three rather than claiming unconditional superiority for any (Štrumbelj's strong dominance claim for Shin does not hold universally). The mean 1X2 overround is 1.067 and the mean Shin z is 0.034.
For a league or window we measure the skewness of the mixture of the per- match two-point distributions. By the law of total cumulants, the aggregate third central moment decomposes as
M₃ = E[m₃ᵢ] + 3·E[σ²ᵢ(μᵢ − μ)] + E[(μᵢ − μ)³],
i.e. a within-match term (the per-match Bernoulli/FLB asymmetry), a covariance term (between match-level variance and mean), and a between-match term (dispersion of match means). This decomposition lets us ask precisely how much of the market's skewness is the mechanical image of the probability distribution versus an emergent property of pooling.
The de-vigged ex-ante object is our primary measurement; the ex-post realised skewness (the skewness of actual returns) is a robustness check that should coincide with it under well-calibrated odds.
The measurement needs, per bet, only a de-vigged probability p, a decimal odd o
and a realised outcome. We therefore factor the analysis behind a canonical data layer
(skewlib/canonical.py; the contract is documented in docs/DATA-SCHEMA.md): a
per-sport adapter maps a raw source to a tidy long form — declaring the outcome
taxonomy (the number of outcomes, and whether a draw exists) and de-vigging the odds —
after which the same selection, decomposition and skew-meter code runs unchanged,
whatever the sport or market width. The football 1X2 path reproduces the legacy numbers
bit-for-bit; the same layer drives the external-validity analysis of §4.9 (tennis,
basketball) without touching the core.
The global ex-ante skewness of the favourite bet is +0.236, against an ex-post realised +0.230 — the implied object reproduces the realised one, confirming aggregate calibration and licensing the cleaner, sampling-noise-free ex-ante measure. The cumulant decomposition is decisive: the within-match term is +102.6% of M₃, the covariance term −2.6%, and the between-match term −0.0%. Market skewness is, to within sampling error, entirely the within- match Bernoulli asymmetry of the win-probability distribution. The same holds conditionally: in every probability bucket the within-match fraction is ≈ 100%, and ex-ante and ex-post skewness track each other tightly along the theoretical identity curve, from +0.51 at the longshot end to −1.22 for strong favourites (Figure 1). The skewness of the betting market is the algebraic image of the favourite–longshot bias — not an emergent feature of aggregation.
Across leagues the ex-ante skewness ranges from 0.10 (Netherlands Eredivisie) to 0.33 (Italian Serie B), with standard deviation 0.06, and is strongly related to how strong favourites typically are. Measured from the odds, corr(mean favourite probability, skewness) = −0.900 — but this is circular, since both quantities are built from the same odds.
To break the circularity we construct competitiveness from a results-only Elo: a single chronological pass over all leagues (win/draw/loss plus goal margin, home advantage), with a rating-difference → (P_H, P_D, P_A) map calibrated on outcomes by multinomial logit (predicted P(H) = 0.444 = realised 0.444). None of the resulting league measures touch market data. Two findings follow. First, the odds- and Elo-based favourite strengths agree across leagues at r = +0.909 [0.83, 0.97]: the odds simply read off the sporting competitive structure (structural efficiency). Second, the law survives the odds-free regressors — skewness vs upset rate r = +0.826 [0.71, 0.91], vs Elo predictive entropy +0.719, vs Elo favourite probability −0.748, vs Elo rating dispersion −0.731 (all bootstrap CIs exclude zero). The attenuation relative to the circular −0.900 is the expected errors-in-variables effect of a noisier proxy, not evidence against the law.
The same holds for the canonical, size-robust competitive-balance indices used in the sports-economics literature, computed from final standings alone — and hence independent of both the odds and our Elo. Following Utt & Fort (2002) we avoid the Gini coefficient (invalid for zero-sum league play) and use the Noll-Scully ratio, the size-normalised HHI* of Owen, Ryan & Weatherston (2007), and the Theil/GE(1) entropy of Borooah & Mangan (2012). All three reproduce the law with the predicted sign — more imbalance, lower skewness: skewness vs Noll-Scully r = −0.625 [−0.83, −0.36], vs HHI* −0.593, vs Theil −0.478. This yields a coherent errors-in-variables ladder, from the (circular) odds at −0.90, through match-level Elo at +0.83, to season-level standings indices at 0.48–0.63: the closer the proxy sits to the per-match win-probability distribution that mechanically generates skewness, the stronger the measured relation. The asymmetry of market returns is inherited from the competitive structure of the league.
Treating (league, season) as the unit of analysis dissolves, by construction, the composition confound that produced an apparent break in earlier pooled analyses (the sample grew from 21 to 37 leagues in 2012). In a 638-observation panel with league fixed effects, the linear year coefficient is +0.00015 per year (cluster-robust SE 0.00045, p = 0.73; 95% CI [−0.0007, +0.0010]) — a twenty-year drift of ≈ +0.003 against a between-league standard deviation of 0.052.
A high p-value, however, is only a failure to reject a trend, not evidence of its absence. We therefore test temporal invariance as an equivalence hypothesis (two one-sided tests, TOST [Schuirmann 1987; Lakens 2017]; Figure 20). Pre-registering the same margin used for the similarity analysis below (§4.8) — half a between-league standard deviation, Δ = 0.026, read as the largest twenty-year drift we would treat as negligible — the slope's 90% confidence interval falls strictly inside [−Δ, +Δ], rejecting the presence of any larger drift at p = 0.006 (TOST). The conclusion is robust to a league-cluster bootstrap (p = 0.005) and to fixing the league basket across all seasons (the balanced 15-league panel, β = −0.0003/yr, p = 0.043), and it tightens monotonically as the margin is relaxed (equivalence at ½ and 1× the between-league SD; inconclusive only at a stringent ¼ SD, i.e. a drift below 0.013 over two decades, which the data can neither establish nor rule out). The skewness is thus not merely not shown to drift — it is statistically equivalent to no drift, to within a fraction of a single league-to-league difference accumulated over the entire record.
A variance decomposition attributes 70% of the total variance to the between-league component (the structural invariant); the within-league standard deviation is 0.034, of which a per-match bootstrap benchmark attributes 0.019 to pure sampling noise, leaving a small (≈ 0.028) real temporal fluctuation that is trendless and mean-reverting — consistent with the white-noise dynamics found in the time-series tests (variance ratios 0.94/0.94/1.00; AR(1) φ = −0.06, p = 0.39). The structural league invariant dominates the temporal component roughly 2:1.
We test directly for within-window regime breaks. A conservative PELT search on each league's season-by-season series finds a single break across all 38 leagues — France 2020 (a −0.064 shift, i.e. COVID) — with no common break year across leagues, and zero breaks for the English top flight (E0: mean 0.165, sd 0.027). This is consistent with, not contrary to, the competitive-balance literature: the regime breaks identified for English football [Lee & Fort 2012; Basini et al. 2023] are tied to shocks (Bosman, the Champions League expansion, the early-2000s revenue divergence) that predate our sample, so 2005–2025 is effectively a single regime. The invariance we document is therefore an intra-regime property of a league-specific baseline.
As a natural experiment, the COVID-19 empty-stadium seasons sharply reduced home advantage (home-win rate 0.447 → 0.417 in 2020). The structural law predicts that weaker home favourites should shift the probability distribution toward parity and raise skewness; we observe a mean within-league shift of +0.42 standard deviations, positive in 21 of 33 leagues. The single exogenous competitiveness shock in twenty years moved skewness in the predicted direction — corroborating the mechanism without contradicting secular invariance.
A second natural experiment sharpens the interpretation by failing to move skewness, as the mechanism predicts it should. The staggered introduction of the Video Assistant Referee (VAR) — adopted by different leagues in 2017/18, 2018/19 and 2019/20 — is an institutional shock that does not alter the dispersion of team strengths, and so should leave the structural asymmetry untouched. A staggered difference-in-differences with league and year fixed effects (treated top divisions against never-treated English and Scottish lower divisions) returns a null effect on ex-ante skewness, β = −0.007 (cluster-robust p = 0.65; −0.14 league standard deviations, confidence interval spanning zero), with equally null effects on the realised favourite win rate and on the market's mean favourite probability, and no discontinuity at adoption in an event-study plot. The contrast with COVID is exactly that of the mechanism: a genuine competitiveness shock (empty stadiums, weaker home advantage) moves skewness; an institutional one that leaves the strength distribution intact does not. Skewness tracks the competitive structure of the sport, not the rules under which it is officiated.
Comparing the market-average odds with the best available price over the same 202,760 matches, taking the best price collapses the overround from 1.067 to 1.009 (the expected return rises from −4.8% to near zero), yet the ex-ante skewness barely moves, +0.236 → +0.254, with a per-match favourite-probability correlation of 0.996. Bookmakers compete on margin (the level of returns), not on the asymmetry (the structure). Pricing and risk-structure are separable.
The same separation holds along the time axis of price formation. On a canonical sample of 34,659 matches (2019/20–2023/24) carrying both opening and closing prices, the asymmetry is already present in the first quote: the ex-ante skewness is +0.248 at the open and +0.249 at the close, correlated at r = 0.998 across leagues, with a mean per-league change of +0.0005. The market does sharpen the price between open and close — the overround falls (1.061 → 1.060) and the favourite's Brier score improves (0.2344 → 0.2330) — but the structural law is already in force at the open (skewness vs favourite probability r = −0.87 at both), and the favourite's probability drifts by only +0.0001 on average. The asymmetry is not manufactured by trading; it is inherited from structure and is in the price before the market refines it.
The over/under-2.5-goals market is a clean two-outcome contract: skewness is a function of a single probability, free of the three-way 1X2 structure. On 148,261 matches the de-vig is well calibrated (over-rate 0.490 = de-vigged p_over 0.492), and the favourite-side bet has ex-ante skewness −0.210 with a within-match fraction of 99.6%, matching the realised −0.217 and the closed-form identity exactly across every probability bucket. The mechanical core is not an artefact of the 1X2 structure.
The mechanical decomposition of §3.3 generalises to every order. A two-point bet has closed-form central moments of all orders, m_k = oᵏ · p(1−p) · [(1−p)^{k−1} + (−1)ᵏ p^{k−1}], and the mixture's k-th central moment follows from the law of total moments, M_k = E_i[Σ_j C(k,j) m_{j,i} dᵢ^{k−j}] with dᵢ = μᵢ − μ. Computing this through the sixth order, the within-match fraction is ≈ 1 at every order (m₂ +1.00, m₃ +1.03, m₄ +1.01, m₅ +1.03, m₆ +1.02): not only the skewness but the entire shape of the implied distribution is the algebraic image of the win-probability distribution, with negligible contribution from how matches are pooled. Globally the implied favourite distribution has variance 0.99, skewness +0.236, excess kurtosis −1.683 (the short-tailed signature of a mixture of Bernoullis), with bootstrap standard errors ≈ 0.001.
The ordered-probit model of §5 predicts not just the third moment but each moment of a league from its mean favourite probability: across the thirty-eight leagues, variance r = +0.99, skewness r = +0.90, excess kurtosis r = +0.89 (Figure 6). The standardised moments (skewness, kurtosis) match in both level and ordering; the raw variance matches in ordering with a multiplicative level offset induced by the overround (real odds o < 1/p). The skewness invariance is thus a corollary of a stronger shape invariance: the whole implied distribution is a single function of league competitiveness.
A distribution-collapse test confirms this directly. Standardising favourite returns within each league (removing location and scale) and comparing leagues pairwise, the median Kolmogorov–Smirnov statistic is 0.474 — the standardised shape genuinely differs across leagues, because skewness varies with the league. But conditioning on competitiveness — comparing each league against the rest within narrow bins of favourite probability — the median KS statistic collapses to 0.059, an 87% reduction (Figure 7). (With samples this large the KS p-value saturates and is uninformative; the statistic itself is the relevant effect size.) Once competitiveness is fixed, league identity adds essentially nothing: the distribution collapses, a stylised fact that the shape is a function of competitiveness and not of the league.
If skewness is structural, is it priced beyond the mechanical bias? The favourite's expected return decomposes exactly into a margin term and a calibration term, r = (p·o − 1) + ((1{win} − p)·o): the first is the vig paid even under perfect calibration, the second the monetised favourite–longshot bias. Globally the favourite returns −4.8%, split into −5.0% margin and +0.15% FLB — the loss is almost entirely the bookmaker's margin, and the favourite side of the longshot bias is small and positive (favourites are mildly underpriced). Decomposing the FLB term into a mechanical level (the global FLB-vs-probability curve applied to each league's probability composition) and a league residual, the residual is uncorrelated with the league's implied skewness (r = +0.11, 95% CI [−0.20, +0.38]); so are the total FLB (−0.04) and the margin (−0.29). There is no league-level skewness premium beyond the mechanical bias: the asymmetry is priced only through the favourite–longshot identity itself, and bookmakers leave no extra, skewness-linked edge — the asymmetry analogue of the margin orthogonality of §4.4.
The bias has a behavioural reading. Modelling the proportionally-devigged implied probability q as the decision weight of the objective probability π through the Tversky–Kahneman weighting w(p) = p^γ/(p^γ + (1−p)^γ)^{1/γ}, the fit gives γ = 0.96 < 1 — the inverse-S that overweights longshots and underweights favourites (calibration: q = 0.10 vs π = 0.09 for longshots, q = 0.71 vs π = 0.74 for favourites). The new point is that this preference parameter is itself an invariant: across seasons γ has mean 0.955, standard deviation 0.020, and no secular trend (β = +0.0003/yr, Δ over 20 years ≈ +0.006), and across leagues it is tightly clustered (mean 0.945, sd 0.040), with only a mild association with competitiveness (r = −0.45). The probability weighting behind the bias is a stable structural constant, not a moving process — the invariance holds on the preference side as well as on the risk side [Snowberg & Wolfers 2010; Barberis & Huang 2008].
The preceding results are most naturally read through a single question: how similar are the return asymmetries of two competitions? We make this operational. For two entities A and B (leagues, eras, markets, or windows) we measure each one's signature — its pooled ex-ante skewness and the mean of its favourite-probability distribution — and define a raw distance D_raw = |skew_A − skew_B| and a competitiveness- conditioned residual distance D_res = |r_A − r_B|, where r_X = skew_X − Ŝ(p̄_X) subtracts the asymmetry that the closed-form law S(·) attributes to each side's competitiveness. Across the thirty-eight leagues the median raw distance is 0.051 — almost exactly the between-league standard deviation — and conditioning on a single competitiveness parameter cuts it to a median residual of 0.023, so one number explains 82% of the between-league asymmetry variance.
The collapse is not yet complete at one parameter, and the reason is informative. The single-parameter residual is not sampling noise: it is a stable, reproducible league trait (its split-half correlation across alternating seasons is r = 0.98), reflecting the curvature of the concave law that a mean cannot carry. Adding the variance of the favourite-probability distribution raises the explained variance to R² = 0.98, and conditioning on the entire favourite-probability distribution drives the residual to the per-league sampling floor (≈ 0.005), with a correlation of 0.997 between observed and predicted league skewness. The full win-probability distribution is thus the minimal sufficient statistic for a league's asymmetry; its mean is 82%-sufficient and its first two moments 98%-sufficient. The same signature is recoverable from minimal inputs — inverse-odds without any de-vig reproduce the league skewness at r = 0.997, and a results-only competitiveness proxy (no market data at all) at r = 0.83 — and a rolling estimate converges to within the between-league spread in about half a season (standard error 0.018 at 400 matches).
Because the sample is large, every pairwise significance test rejects equality, so we report a calibrated equivalence verdict instead. Fixing a margin of half a between- league standard deviation (Δ = 0.026) and applying two one-sided tests, the English and Spanish top flights — and, strikingly, the English first and fourth tiers, whose raw skewness differs by 0.13 — are equivalent in conditioned asymmetry, while the Dutch top flight and the Italian second tier are not. The similarity of asymmetries is therefore, to first order, the similarity of competitiveness: measurable with one parameter, made exact by the full probability distribution, and verdicted by an equivalence test rather than a null-rejection.
Our evidence so far is from football and, mostly, the three-outcome 1X2 contract. To separate the structural claim from anything idiosyncratic to that sport or that market, we port the analysis — through the canonical layer of §3.4, with no change to the core — to two further sports whose odds come from independent providers. Tennis (tennis-data.co.uk, ATP and WTA, 62,865 matches) is a two-outcome match-odds market with no draw; basketball (sportsbookreviewsonline.com, NBA, sixteen seasons, 19,621 games) is a two-outcome moneyline market, also with no draw.
In both, the de-vig stays calibrated out of sample (mean favourite probability 0.688 vs realised 0.692 in tennis; 0.694 vs 0.685 in basketball, the residual gap being the favourite–longshot bias itself), so the implied distribution is trustworthy out of domain. And in both the structural law reappears at full strength: the favourite bet is most negative where the competition is most lopsided — corr(skewness, competitiveness) = −1.00 (ATP) and −0.98 (WTA) across tournament tiers, and −0.95 across NBA seasons, against −0.90 in football — while the underdog is lottery-like at +2.31 (tennis) and +2.61 (basketball), all but identical to football's +2.35. Placed on a single competitiveness axis (Figure 19), the three sports trace one curve: football at the balanced end, tennis and basketball overlapping in the more lopsided regime, the favourite falling and the underdog rising with imbalance throughout. The asymmetry is not an artefact of the 1X2 contract, of the football-data source, or of association football — it is a property of the sport as a competitive system, exactly as the mechanism of §5 predicts.
The results cohere under one principle. The skewness of a fixed-odds bet is the Bernoulli asymmetry of its win probability; the market's aggregate skewness is the average of these over the matches on offer, with negligible contribution from how the matches are mixed (§4.1). A league's typical favourite strength — its competitiveness — therefore determines its skewness almost deterministically (§4.2). Because competitiveness is a slow-moving property of a sporting competition (its scale of change is decades, governed by the distribution of club strengths, promotion/relegation and the like), the skewness it pins down is a temporal invariant (§4.3). What looks like a question about market dynamics is answered by a property of the sport: the cancellation of the two tails (weak favourites contributing right-skew, strong favourites left-skew) is stable because the underlying competitive distribution is stable. Bookmaker margin shifts the level of returns but not this asymmetry (§4.4), and the principle is market- agnostic (§4.5).
This is not merely a verbal account: the law can be derived. Following the ordered-probit tradition for football outcomes [Goddard & Asimakopoulos 2004; Koning 2000], let team strengths be Gaussian with within-league dispersion σ_L, so a match's strength gap is d ~ N(0, 2σ_L²); a latent margin y* = d + h + ε with home advantage h and standard-normal noise generates (away, draw, home) through a pair of cut-offs ±c, and the favourite probability is p = max(P_H, P_D, P_A). Under fair odds the favourite bet has zero mean, so a league's pooled skewness is the closed functional S(σ_L) = E[m₃(p)] / E[σ²(p)]^{3/2} with σ²(p) = (1 − p)/p and m₃(p) = (1 − p)(1 − 2p)/p² — a single-peaked (concave) function of the lone competitiveness parameter σ_L. Calibrating (h, c, σ) once to the pooled marginal rates (home 0.444, draw 0.264, mean favourite probability 0.499) gives h = 0.22, c = 0.37, and the resulting curve predicts each league's third moment from its first moment (mean favourite probability) at r = +0.90, RMSE 0.024 — less than half the between-league standard deviation. The thirty-eight leagues lie on the derived curve (Figure 5): the skewness–competitiveness law is an analytic consequence of a strength model plus the favourite–longshot identity, not a free fit.
The law in closed form (Figure 10). S(σ_L) need not be simulated. Its two expectations are one-dimensional Gaussian integrals in the strength gap d, which we evaluate by adaptive quadrature, partitioning at the gap values where the favourite identity switches (the kinks of p_fav). The closed-form curve reproduces the Monte-Carlo estimate to a maximum absolute difference of 0.0015 — the Monte-Carlo noise floor — over the whole σ_L range, removing all simulation error. The balanced limit is fully analytic: as σ_L → 0, S(σ_L) → (1 − 2p₀)/√(p₀(1 − p₀)) with p₀ = Φ(h − c), the per-match identity evaluated at the equilibrium favourite (p₀ = 0.439, S₀ = +0.245 here), and the leading correction S₂ = +8.4 σ_L² is positive — skewness rises as a league departs from perfect balance. The curve is not monotone: it peaks at σ* = 0.12 (S_max = +0.30, mean favourite probability 0.45) and declines thereafter, crossing zero only when one side becomes overwhelmingly favoured. Because p_fav is piecewise-smooth (kinked where the favourite changes identity), S(σ_L) is C^∞ but not globally analytic — its Taylor series about balance has a finite radius, so the closed form is the integral itself rather than an elementary series. Evaluated exactly, the curve still places the thirty-eight leagues at r = +0.90.
Why the within-match term dominates (tail cancellation). The empirical finding that M_k ≈ E[m_{k,i}] at every order has a simple cause. Under fair odds the favourite bet has mean exactly zero, so every deviation dᵢ = μᵢ − μ vanishes and the law-of-total-moments expansion collapses to M_k = E[m_{k,i}] identically — every cross term carries a factor of some dᵢ. Real odds carry an overround, so the means are slightly negative rather than zero, but uniformly so: the dᵢ are small and tightly clustered, and the covariance and between-match terms, being O(E[dᵢ]) and O(E[dᵢ³]), stay near zero (−2.6% and −0.0% of M₃). The mixing of matches cannot generate asymmetry because the matches barely differ in mean; all the shape comes from within each two-point bet. A complementary cancellation explains why the level of skewness is insensitive to league composition while the variance is not: variance is monotone in favourite strength (corr −0.90), so reweighting matches moves it, whereas the two skewness tails — weak favourites contributing positive skew, strong favourites negative — offset, leaving aggregate skewness (corr with strength ≈ −0.2) and its composition nearly decoupled. Stability of the competitive distribution therefore buys stability of skewness for free.
Model independence (Figure 14). The law is a property of the two-point mixture, not of the ordered-probit form used to derive it. We regenerate each league's match-level outcome probabilities from a battery of independent generators and ask whether each reproduces the cross-league skewness ordering and lands on the derived curve. A Poisson goals model (attack/defence plus home effect, fit per league-season over 617 league-seasons) recovers competitiveness at r = +0.97 and the skewness ordering at r = +0.93; a Dixon-Coles model adding low-score dependence (mean ρ = −0.05) raises the implied level toward the market (+0.20 against the market's +0.22) at r = +0.87; a Bradley-Terry-Davidson model — multiplicative team strengths with a tie parameter, fit by paired comparison and using no goals at all — reproduces it at r = +0.84; and the results-only Elo of §4.2, a fully odds-free generator, sits on the curve at r = +0.96. All five families — latent-margin probit, Poisson, Dixon-Coles, Bradley-Terry-Davidson and the results Elo — together with the market, fall on a single curve (r on the derived curve from +0.85 to +0.96 against +0.90 for the market). Models built to deviate do not: because football goals are very nearly independent Poisson (median home–away goal covariance −0.07, negligible overdispersion), both a bivariate-Poisson generator (positive goal correlation) and a negative-binomial generator (overdispersion) collapse back to the Poisson result. The skewness–competitiveness law is the geometry of the two-point mixture over the league's competitiveness distribution, not an artefact of any one functional form.
The central findings are insensitive to analytical choices. The global ex-ante skewness is 0.224 / 0.236 / 0.263 under power / Shin / multiplicative de-vigging, while the cross-league law is essentially unchanged (corr −0.906 / −0.900 / −0.891). The ordering is itself a sanity check: the multiplicative method, which removes the margin proportionally and so under-corrects the longshot side, returns the highest skewness, exactly as expected when a de-vig ignores the favourite–longshot structure [Clarke, Kovalchik & Ingram 2017]. Stationarity is confirmed by both ADF (p < 0.001) and KPSS (p > 0.10, the test's tabulated maximum — stationarity is not rejected) across window sizes from 500 to 3000 matches and under league-demeaning. The apparent short-run persistence in overlapping windows (ACF₁ = 0.74) is entirely a window-overlap artefact: with disjoint windows the series is white noise (Ljung-Box p = 0.70, ACF₁ = −0.06). The per-match bootstrap benchmark (§4.3) quantifies how much within-league variation is mere sampling noise. Ex-ante and ex-post measures agree throughout.
Three adversarial checks close the robustness case. First, the de-vig is trustworthy and the result does not depend on it: the favourite probability is near-perfectly calibrated against realised outcomes (Brier reliability term 0.000 globally, with a between-league standard deviation of 0.0003 and no badly calibrated league or season), and the aggregate skewness ranges only +0.224 to +0.263 across Shin, multiplicative and power de-vigs, the best-price book, and a multi-book consensus (Figure 12). Second, composition is not driving the invariance: rebuilding the global series from only the fifteen leagues present in all twenty-one seasons leaves the trend flat (β = −0.0001/yr, twenty-year change −0.003, level +0.243 ± 0.014; Figure 13), so the absence of drift is not an artefact of a changing league basket. Third, the headline numbers carry honest confidence intervals: a block-bootstrap that resamples whole seasons (respecting within-year dependence) gives global skewness +0.236 [+0.232, +0.239] and a structural-law correlation of −0.90 [−0.92, −0.88] — both comfortably excluding the null.
The derivation also does not lean on the Gaussian strength assumption (Figure 11). A match's strength gap is the difference d = rᵢ − rⱼ of two independent team strengths, which is symmetric about zero for any identically distributed strength law — so asymmetry in the strengths themselves cannot bias the result; only tail weight can. Replacing the Normal strengths with Student-t (raising the excess kurtosis of d from ~0 to as much as 36), with a skew-normal (±α), or with a uniform law, and comparing curves at matched competitiveness (mean favourite probability), the skewness–competitiveness relationship shifts by at most |ΔS| = 0.03 — smaller than the between-league standard deviation of 0.05. The skew-normal curves collapse onto the Gaussian almost exactly, precisely as the symmetry argument predicts, and the residual movement scales with the kurtosis of d, not its skew. The law is the geometry of the two-point mixture, not an artefact of normality.
Finally, we guard against a specific confound: if the favourite–longshot bias itself drifted over the sample — recent work reports a weakening of the bias in European data [Angelini & De Angelis 2019] — a moving bias could masquerade as skewness invariance. It does not. Over 2005–2025 the bias barometer (the underdog return) shows no significant trend (correlation with year +0.27, 95% CI [−0.23, +0.67]; a faint, non-significant tendency in the reported direction), the favourite−underdog return spread is flat (corr −0.02), and the favourite's annual calibration error stays within [−0.004, +0.012]. Ex-ante and ex-post skewness coincide every year (mean absolute difference 0.015). The invariance is therefore not an artefact of a time-varying bias; and because skewness is mechanically set by the probability distribution, it is in any case robust to mild calibration drift.
Our results recast a familiar object. The favourite–longshot bias is usually read as a statement about prices — a pricing anomaly explained by skewness preference. We show that, viewed as the skewness of the market's return distribution, the phenomenon is better described as a structural invariant of the sport that prices faithfully transcribe: the odds-implied and results-based measures of competitive structure agree at r = 0.91, and the resulting skewness is flat across two decades. The market is, in this dimension, efficient and stable — it neither learns nor drifts, because there is nothing to learn: the asymmetry is exogenous to pricing.
This contrasts with financial markets, where return skewness is a conditional, time-varying object actively priced as a risk factor [Kraus & Litzenberger 1976; Harvey & Siddique 2000] and rationalised through probability weighting [Barberis & Huang 2008; Boyer, Mitton & Vorkink 2010]. In betting markets the terminal payoff is exogenous and the skewness is mechanically tied to a slow sporting primitive, so the same statistical object behaves as a constant rather than a process. The betting market thus isolates the "supply" side of skewness — set by the event generator — from the "demand" side studied in asset pricing.
Limitations. Competitiveness is a latent construct; our Elo proxy, though odds- free, is noisier than the odds themselves, attenuating the estimated law. Team identities are matched by name across a global pool, which may conflate rare homonyms. The dataset is a normalised mirror; we corroborate the key results on the primary football-data.co.uk source directly (the price-discovery and pre-2005 analyses above use it). On the regime question, extending the record back to 2000 with a single continuous bookmaker (William Hill) shows no break at our 2005 cutoff and a per-league baseline ordering preserved across the boundary (r = 0.76), with at most a faint, endpoint-sensitive drift in level (the modern era ≈ 0.018 above 2000–2004, p ≈ 0.03–0.10) — the modern competitive regime already obtains by 2000. The institutional regime breaks the literature identifies, however, are earlier still (Bosman 1995, the Champions League expansion of 1994/95), and 1X2 odds do not exist before ≈ 2000, so the prediction that the baseline itself shifts across those 1990s shocks remains beyond the reach of betting data.
The structural reading also makes a portable prediction, which §4.9 confirms: ported unchanged to tennis and basketball — different sports, two-outcome markets with no draw, independent odds sources — the same law reappears (the favourite's skewness falling with competitiveness, the underdog lottery-like near +2.3 to +2.6, the de-vig still calibrated out of domain). The asymmetry is therefore a property of the sport as a competitive system, not an artefact of the 1X2 contract or of football — which is what a mechanism rooted in the win-probability distribution, rather than in pricing, predicts.
The skewness of football betting returns is a structural invariant. It is, to within sampling error, the algebraic image of the win-probability distribution (the favourite–longshot bias) — and not merely its third moment but its entire shape, which collapses across leagues once competitiveness is held fixed; its league-level value is fixed by sporting competitiveness — a relation that survives an odds-free measure of competitiveness — and it shows no secular drift within the modern competitive regime that spans our window. Bookmaker margin is orthogonal to it, and the identity holds in a second, binary market. No league-level premium is priced beyond the mechanical bias, and the preference that underlies that bias — the probability weighting of cumulative prospect theory — is itself a stable structural constant, showing no drift over twenty years. The market's risk asymmetry is inherited from the sport, not produced by the pricing of it.
Ethics. Not applicable. This study uses only publicly available aggregate match results and bookmaker odds; it involves no human participants, personal data, or animal subjects.
Data accessibility. All derived data needed to reproduce every figure and
statistic in this article are openly available: the per-league and per-season
aggregate tables (study/outputs/*.csv), the consolidated results artifact
(site/src/data/findings.json), and the figures, all with a data dictionary
(study/outputs/README.md). These are archived at Zenodo
(doi:10.5281/zenodo.20822121 — concept DOI, resolving to the latest version) and
version-controlled at the
repository below. The underlying raw match-and-odds file is sourced from
football-data.co.uk; its terms of use restrict redistribution, so it is not
deposited. It is regenerated deterministically by analysis/00_fetch_data.py,
which verifies the exact analysis snapshot against the frozen SHA256
(6905ca5324df…, recorded in study/data/PROVENANCE.json). The provided derived
data are sufficient to replicate all reported findings without the raw file.
Code accessibility. All analysis code is open source (Apache-2.0) at
https://github.com/cold-code-labs/skew-dynamic. The complete pipeline reproduces
every number, figure, and the evidence ledger from one command (cd study && ./run.sh), with a pinned environment (requirements.txt / requirements.lock,
Python in .python-version) and an automatic result-drift audit
(build_lineage.py --check). Each headline number is pinned to the exact code
version that produced it via a versioned lineage ledger and git evidence tags.
Competing interests. The author declares no competing interests.
Funding. This work received no external funding; it was conducted by Cold Code Labs.
Authors' contributions. V.A.: conceptualization, methodology, software, formal analysis, data curation, writing — original draft, writing — review and editing, visualization.
Use of AI. The author used an AI coding assistant (Anthropic Claude) to help implement and review analysis code and to copy-edit prose. All study design, methodological choices, results, and conclusions are the author's; all AI-assisted code and text were verified against the data and the result-drift audit.
- Figure 1. Favourite–longshot bias: ex-ante and ex-post skewness against favourite probability, on the theoretical identity curve (1 − 2p)/√(p(1 − p)).
- Figure 2. The structural law: league ex-ante skewness against odds-free competitiveness (Elo upset rate), r = +0.83.
- Figure 3. Decomposition of the aggregate third moment into within-match, covariance and between-match components.
- Figure 4. League×season panel: no secular trend within the modern regime.
- Figure 5. The derived law: leagues lie on the ordered-probit curve relating mean favourite probability to skewness (r = +0.90).
- Figure 6. Shape invariance: each league's skewness and excess kurtosis lie on the derived ordered-probit curve as a function of mean favourite probability (r = +0.90 and +0.89).
- Figure 7. Distribution collapse: standardised favourite-return ECDFs differ across leagues (left), but collapse onto one another within a fixed band of competitiveness (right).
- Figure 8. Return decomposition: the mechanical FLB curve by favourite probability (left), and total FLB vs league residual against implied skewness — no league-level premium beyond the mechanical level (right).
- Figure 9. Invariant preference: the fitted Tversky–Kahneman probability weighting (left) and γ by season, flat within the between-league band (right).
- Figure 10. Closed form: S(σ_L) by exact quadrature reproduces the Monte-Carlo curve (left, max |Δ| = 0.0015) and, evaluated exactly, places the leagues on the closed curve (right, r = +0.90).
- Figure 11. Strength-law robustness: the skewness–competitiveness curve under Normal, Student-t (ν = 3, 5), skew-normal and uniform strengths, near-coincident when reparametrised by mean favourite probability (max |ΔS| = 0.03).
- Figure 12. De-vig reliability: the favourite's reliability diagram (predicted vs realised win frequency) and the calibration-error term by league — small and stable, so the implied skewness is not a de-vig artefact.
- Figure 13. Balanced panel: the global skewness series from the fifteen always-present leagues lies flat against the full-basket series — no drift once composition is held fixed.
- Figure 14. Model independence: five independent outcome generators (latent-margin probit, Poisson, Dixon-Coles, Bradley-Terry-Davidson, results-only Elo) and the market all fall on one skewness–competitiveness curve (left), each reproducing the cross-league skewness ordering against the market (right).
- Figure 15. Price discovery: the ex-ante skewness at the opening price equals that at the close (r = 0.998 across leagues, left), while the margin shrinks from open to close (right) — the asymmetry is in the first quote, not built by trading.
- Figure 16. VAR natural experiment: an event study shows no jump in skewness at VAR adoption (left), and the difference-in-differences effect is null in league standard deviations against the COVID competitiveness shock of §4.3 (right).
- Figure 17. Pre-2005 regime: the league-demeaned global skewness series shows no break at the 2005 cutoff (left), and the per-league baseline is preserved between 2000–2004 and the modern era (r = 0.76, right).
- Figure 18. Similarity of asymmetries: the pairwise raw |Δskew| matrix across leagues (left) collapses once competitiveness is conditioned out (middle), and the sufficiency ladder shows one parameter explains 82%, the first two moments 98%, and the full win-probability distribution is sufficient (right).
- Figure 19. External validity across sports: ex-ante skewness against competitiveness (mean favourite probability) for the favourite bet (left) and the underdog bet (right), pooling football leagues, tennis tiers (ATP+WTA) and basketball seasons (NBA). The three sports — three two-outcome/three-outcome markets, three independent odds sources — trace one descending (favourite) and one rising (underdog) curve.
- Figure 20. Temporal invariance as equivalence: the twenty-year drift in ex-ante skewness (slope × span) with its 90% confidence interval, for the full panel (cluster-robust and league-bootstrap) and the balanced 15-league panel, against the pre-registered equivalence band ±Δ (half a between-league SD). Every interval lies inside the band — positive evidence of no drift, not a mere failure to reject β = 0.
Listed author–date (valid for RSOS format-free initial submission). Final-stage copy-edit: convert to numbered Vancouver style and add DOIs.
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