A Hidden Invariant in International Football: Spectral Gap Stability of the Win–Draw–Loss Markov Chain (1902–2026)

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Abstract

We model sequences of international football match outcomes (win, draw, loss) as a first-order Markov chain and study the evolution of its spectral properties over 120 years of data. Despite significant secular declines in the diagonal transition probabilities — teams have become measurably less "streaky" since the early twentieth century — the spectral gap of the 3×3 transition matrix remains effectively constant at 0.37 ± 0.02 across all decades from the 1910s to the 2020s. This implies a fixed mixing time of approximately 7 matches: the number of games required for a team's outcome distribution to forget its initial state has not changed in over a century. The result is robust to bootstrap resampling, holds across competitive and friendly fixtures separately, and persists across confederations. We argue that this spectral invariance constitutes a hidden structural property of international football — surface statistics change, but the chain's rate of convergence to equilibrium does not.

1. Introduction

A recurring question in sports analytics is whether football has become more competitive over time. Conventional indicators — frequency of upsets, distribution of goals, concentration of tournament winners — suggest that the modern game is less predictable than its predecessors (Dobson & Goddard, 2001; Castellano, 2018). Draws are more common than a century ago, dominant winning streaks are rarer, and the global expansion of the sport has deepened the pool of competitive nations. These observations invite a natural follow-up: has the dynamical structure of outcomes changed alongside the surface statistics?

We approach this question through the lens of Markov chain theory. Each team's sequence of international results — win (W), draw (D), or loss (L) — is modeled as a realization of a first-order Markov chain on three states. The transition matrix encodes conditional outcome probabilities (e.g., the probability of winning given that the previous match was a loss), and its spectral properties determine how quickly the chain "forgets" its initial state. The spectral gap — the difference between the largest and second-largest eigenvalue magnitudes — controls the mixing time, i.e., the number of steps required for the chain's distribution to converge to stationarity (Levin, Peres & Wilmer, 2009).

The central finding of this paper is an invariance result. While individual transition probabilities have changed substantially — P(W→W) and P(L→L) have both declined significantly over 120 years, consistent with the narrative of increasing competitiveness — the spectral gap has not. It has remained flat at approximately 0.37 from the 1910s through the 2020s, implying a constant mixing time of roughly 7 matches. The transition structure compensates: as diagonal entries fall, off-diagonal entries redistribute in a way that preserves the second eigenvalue's magnitude. Football's "memory" — how many matches it takes for the outcome process to become independent of its starting state — appears to be a structural constant of the international game.

2. Methods

2.1 Data

We use the publicly available international football results dataset compiled by Martj42 (2024), which records every official men's international match from 1872 to the present. The analysis window spans 1902–2026, covering 49,076 matches across 333 national teams and yielding 97,653 outcome transitions. Each match produces a pair of outcomes: one for the home team and one for the away team, each classified as W, D, or L. A team's outcome sequence is the chronological series of its results across all international fixtures.

2.2 Markov chain formulation

We model the outcome sequence as a first-order, time-homogeneous Markov chain on the state space (\mathcal{S} = {W, D, L}). The 3×3 transition matrix (P) has entries

[P_{ij} = \Pr(X_{n+1} = j \mid X_n = i), \quad i,j \in \mathcal{S}]

estimated by pooling all consecutive outcome pairs across all teams within a given time window. Rows are normalized to sum to 1. Because each row contains positive entries (every transition is observed), the chain is irreducible and aperiodic, hence ergodic with a unique stationary distribution (\pi).

2.3 Spectral gap and mixing time

The transition matrix (P) has eigenvalues (1 = \lambda_1 \geq |\lambda_2| \geq |\lambda_3|). The spectral gap is defined as

[\gamma = 1 - |\lambda_2|]

and the mixing time (the number of steps until the total variation distance to stationarity falls below (\varepsilon)) is bounded by

[t_{\text{mix}}(\varepsilon) \leq \frac{\ln(|\mathcal{S}|/\varepsilon)}{\gamma}]

We report (t_{\text{mix}}) with (\varepsilon = 0.25), giving an upper bound of (\ln(3/0.25)/\gamma = \ln(12)/\gamma).

For a 3×3 matrix, eigenvalues are computed analytically by solving the characteristic polynomial via Cardano's formula. The entire pipeline — data parsing, transition counting, eigenvalue computation, bootstrap resampling — is implemented in standard-library Python with no external numerical dependencies.

2.4 Trend analysis

To test whether the spectral gap exhibits a temporal trend, we compute decade-level estimates (1910s through 2020s, twelve decades) and apply the Spearman rank correlation between decade midpoint and spectral gap. The same test is applied to the diagonal entries P(W→W) and P(L→L) to verify that individual transition probabilities do trend over time.

2.5 Bootstrap confidence intervals

For each decade, we construct 95% confidence intervals for the spectral gap by resampling teams with replacement (1,000 bootstrap replicates). In each replicate, the transition matrix is re-estimated from the pooled transitions of the resampled team set, and the spectral gap is recomputed. The 2.5th and 97.5th percentiles of the bootstrap distribution define the confidence interval.

2.6 Subgroup analyses

We stratify results by match type (competitive vs. friendly, as labeled in the source data) and by confederation (UEFA, CONMEBOL, AFC, CAF; post-1960 only, to ensure adequate sample sizes). The same Markov chain estimation and spectral analysis is applied within each subgroup.

3. Results

3.1 Global transition matrix

Pooling all 97,653 transitions from 1902 to 2026 yields the following transition matrix:

The stationary distribution is (\pi = (0.387,; 0.228,; 0.385)), indicating that wins and losses are nearly equally likely at equilibrium, with draws occurring roughly 23% of the time.

The spectral gap of this matrix is (\gamma = 0.367), corresponding to a mixing time upper bound of (t_{\text{mix}} = 6.8) matches.

Several features of (P) are noteworthy. The diagonal dominance is moderate: the most probable next outcome is the same outcome, but only barely — P(W→W) = 0.445 and P(L→L) = 0.461 represent mild persistence, not strong streakiness. The draw state has the weakest self-transition (0.242), consistent with draws being unstable equilibria that teams tend to exit.

3.2 Decade-by-decade evolution

Table 1 presents the spectral gap, mixing time, diagonal transition probabilities, draw frequency, and sample size for each decade.

Table 1. Spectral gap and transition properties by decade.

The spectral gap ranges from 0.356 (1930s) to 0.380 (1910s, 1950s, 2020s) — a total variation of 0.024 over twelve decades. The mixing time correspondingly oscillates between 6.5 and 7.0 matches. By contrast, P(W→W) ranges from 0.432 to 0.525, and draw frequency rises from 15.0% to 26.3% before partially receding. The surface statistics of football have changed; its spectral structure has not.

3.3 Trend tests

Spearman rank correlations confirm this divergence:

Table 2. Spearman rank correlation with decade (n = 12 decades).

The diagonal entries — the "stickiness" of wins and losses — have declined significantly over time. Wins are less likely to follow wins; losses are less likely to follow losses. The game has indeed become less streaky. Yet the spectral gap shows no significant trend (ρ = 0.294, p = 0.331). The chain's convergence rate is decoupled from the individual transition probabilities that compose it.

3.4 Bootstrap confidence intervals

Bootstrap 95% confidence intervals for the spectral gap overlap substantially across all decades, consistently spanning the approximate range 0.35–0.40. No decade's interval is disjoint from any other's. This confirms that the observed stability is not an artifact of point estimation: even accounting for sampling variability (particularly relevant for earlier decades with fewer transitions), the spectral gap cannot be distinguished across eras.

3.5 Competitive vs. friendly matches

Stratifying by match type yields a modest but interpretable difference:

Table 3. Spectral properties by match type.

Competitive matches (World Cup qualifiers, continental championships, etc.) show a slightly larger spectral gap — faster mixing — than friendlies. This is consistent with the interpretation that competitive fixtures, where stakes enforce effort and tactical discipline, produce a tighter, less autocorrelated outcome process. The difference, however, is small (Δγ = 0.021), and both values fall within the historical range observed across decades.

3.6 Confederation analysis

Restricting to post-1960 data to ensure adequate representation, we estimate spectral gaps by confederation:

Table 4. Spectral gap by confederation (post-1960).

UEFA, the most commercially developed confederation, has the largest spectral gap (fastest mixing), while CAF, where resource disparities between teams are widest, has the smallest. The range across confederations (0.025) is comparable to the range across decades (0.024), and all values remain close to the global mean of 0.367. The invariance is not merely temporal; it extends across the sport's geographic and organizational subdivisions.

4. Discussion

4.1 The compensation mechanism

The central puzzle is why the spectral gap remains constant when its constituent transition probabilities do not. The answer lies in the structure of the 3×3 stochastic matrix. The spectral gap depends on (|\lambda_2|), which is determined by the full matrix — not just the diagonal. As P(W→W) and P(L→L) decline (the diagonal shrinks), the off-diagonal mass redistributes. In particular, the increased draw frequency and the greater symmetry of off-diagonal transitions in the modern era offset the reduced diagonal persistence. The second eigenvalue's magnitude is preserved because the net effect on the characteristic polynomial's non-trivial roots is negligible.

More precisely, consider the decomposition (P = \pi \mathbf{1}^T + \lambda_2 \mathbf{v}_2 \mathbf{w}_2^T + \lambda_3 \mathbf{v}_3 \mathbf{w}_3^T), where (\mathbf{v}_i) and (\mathbf{w}_i) are right and left eigenvectors. The stationary component (\pi \mathbf{1}^T) absorbs the secular changes (more draws, fewer streaks), while the transient components — which govern the rate of convergence — remain stable. This is not a tautology: there is no a priori reason for (|\lambda_2|) to be invariant under changes to (P). It is an empirical property of how football's outcome structure has evolved.

4.2 Interpretation of mixing time

The mixing time of ~7 matches has a concrete interpretation: after 7 games, a team's current result is essentially independent of its result 7 games ago. This defines a natural "memory horizon" for the international game. Streaks shorter than 7 matches are partially explained by momentum (autocorrelation in the Markov chain); streaks longer than 7 matches require an explanation beyond the baseline stochastic model — e.g., genuine differences in team quality, coaching changes, or generational talent.

That this memory horizon has been stable for over a century is striking. It suggests that while the content of football has changed — tactics, fitness, globalization, professionalization — the temporal structure of its outcome process has not. The game's dynamics operate within a fixed regime.

4.3 Limitations

Several caveats are appropriate. First, the first-order Markov assumption is a simplification. Higher-order dependencies may exist, and the true outcome process is influenced by covariates (opponent strength, venue, match importance) that the model ignores. The spectral gap characterizes the pooled, unconditional chain and should not be interpreted as a universal law governing individual team trajectories.

Second, the mixing time is an upper bound derived from the spectral gap. The actual convergence may be faster — the bound is tight only when the initial distribution is maximally misaligned with the stationary distribution.

Third, the early decades (1910s–1940s) have substantially fewer transitions than the later ones, resulting in wider confidence intervals. The invariance claim is strongest for the post-1950 era, where sample sizes exceed 3,000 transitions per decade.

Finally, the pooling of all teams within a decade treats the outcome process as exchangeable across teams. In reality, top-ranked and bottom-ranked teams have different transition matrices. The pooled chain represents a population average, and the spectral gap's stability may partially reflect the averaging itself. Disaggregated analyses by team tier would be a natural extension.

4.4 Implications

The invariance of the spectral gap has implications for several areas of football analytics. For competitive balance research, it suggests that traditional metrics (e.g., win concentration, Gini coefficients of points) may capture surface variation without reflecting deeper structural changes. For prediction models, it implies that the autocorrelation structure of outcomes has been stationary, which simplifies time-series modeling. And for the philosophy of sport, it poses an intriguing question: is the ~7-match memory horizon a contingent feature of football's rules and format, or does it reflect something more general about competitive dynamics in team sports?

5. Conclusion

We have shown that the spectral gap of the win–draw–loss Markov chain in international football is a hidden invariant, stable at approximately 0.37 across twelve decades, multiple match types, and all major confederations. Individual transition probabilities have changed — teams are measurably less streaky today than a century ago — but the chain's mixing rate has not. Football's memory is ~7 matches, and it has been ~7 matches for as long as we can measure. The surface statistics of the game evolve; its spectral skeleton does not.

References

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