Abstract

We revisit our previous bidimensional extension of the Zermelo model for chess tournaments with draws (Forlano, 2026), where each player is characterized by both a strength parameter and a draw propensity parameter. While the model provides a flexible and expressive probabilistic framework, it introduces \(2N\) free parameters, which may lead to overfitting in sparse settings such as Swiss-system tournaments.

In this work, we propose a reduced-parameter formulation in which the draw propensity is constrained to depend linearly on player strength. This yields a model with only one degree of freedom per player plus a small number of global parameters. We show that, for a specific choice of the coupling coefficient, the model reduces to the classical Davidson formulation.

We then focus on a first-order linearization around Elo ratings, leading to a symmetric Laplacian system that admits a closed-form solution. This yields a computationally efficient approximation of the full maximum likelihood estimator, while preserving its main structural features.

The proposed method also reveals a structural property of Swiss tournaments: deterministic pairing rules induce systematic biases in the comparison graph, which degrade the reliability of score-based rankings.

This phenomenon is consistent with recent findings by Sauer et al. (2021), who show that alternative pairing mechanisms improve ranking accuracy.

In contrast, our approach leaves the pairing unchanged and provides a graph-based correction at the ranking level.

Introduction

Ranking players in tournaments with draws is a classical problem in paired comparison theory. Models such as the Zermelo model and its extensions provide a probabilistic foundation, but typically require solving nonlinear systems.

In previous work (Forlano, 2026), we considered a bidimensional extension in which each player \(i\) is associated with:

• a strength parameter \(r_i\),
• a draw parameter \(\sigma_i\).

While expressive, this approach introduces \(2N\) parameters, which may be problematic in sparse comparison graphs such as those arising in Swiss-system tournaments.

In this paper, we adopt a reduced-parameter approach: we reduce the dimensionality of the model by coupling the draw parameter to the strength parameter, and we develop a linearized formulation that leads to a closed-form solution.

A Reduced-Parameter Model

We assume that each player is described by a single latent strength parameter \(r_i\), and define: \[ u_i = e^{r_i} \]

Instead of treating the draw propensity as an independent parameter, we impose the constraint:

\[ \sigma_i = \alpha + \beta r_i \]

where:

• \( \alpha \) is a global parameter,
• \( \beta \) controls the coupling between strength and draw tendency.

This reduces the model to:

• \(N\) player parameters \( r_i \),
• plus global parameters \( \alpha \), \( \beta \).

For a pair of players \( (i,j) \), define:

\[ Z_{ij} = u_i + u_j + e^{\sigma_i + \sigma_j} \]

The outcome probabilities are:

\[ P_{ij}^W = \frac{u_i}{Z_{ij}}, \quad P_{ij}^L = \frac{u_j}{Z_{ij}}, \quad P_{ij}^D = \frac{e^{\sigma_i + \sigma_j}}{Z_{ij}} \]

This is a generalized Davidson-type model where the draw component depends on player strengths. We also define:

\[ p_{ij} = P_{ij}^W, \quad q_{ij} = P_{ij}^L, \quad t_{ij} = P_{ij}^D \]

For the specific choice:

\[ \beta = \tfrac{1}{2} \]

we obtain:

\[ \sigma_i + \sigma_j = 2\alpha + \tfrac{1}{2}(r_i + r_j) \]

so that:

\[ e^{\sigma_i + \sigma_j} = e^{2\alpha} \sqrt{u_i u_j} \]

This recovers exactly the classical Davidson model with \( \eta = e^{2\alpha} \). Therefore the reduced model contains Davidson as a special case but allows more general couplings through \( \beta \).

Although different values of \( \beta \) are admissible and may be explored in future work, in this paper we focus exclusively on the case \( \beta = 1/2 \) due to its structural properties that will be highlighted below.

In the Davidson formulation, the probability of a draw depends on a global parameter \(\alpha\), through \[ \eta = \exp(2\alpha), \]

which controls the overall propensity of draws.

In the Laplacian Davidson implementation, \(\alpha\) is not estimated jointly with player strengths. Instead, it is calibrated from the empirical draw rate of the tournament.

Let \[ \delta = \frac{\text{total number of draws}}{\text{total number of games}} \] denote the observed draw fraction.

To relate \(\delta\) to \(\alpha\), we consider the homogeneous Davidson model, i.e. a model in which all players share the same draw propensity. In this case, the draw probability between two players of equal strength is \[ P^D = \frac{\eta}{2+\eta}. \]

Since Swiss tournaments predominantly pair players of similar strength, we approximate the global draw fraction by this quantity: \[ \delta \approx \frac{\eta}{2+\eta}. \]

Solving for \(\eta\) yields \[ \eta = \frac{2\delta}{1-\delta}, \qquad \alpha = \frac{1}{2} \ln \left(\frac{2\delta}{1-\delta}\right). \]

This value is then used as a fixed global parameter in the linearization around the expansion point.

This calibration should be interpreted as a mean-field approximation: the heterogeneous interaction structure of the tournament is replaced by an effective interaction between players of comparable strength. In practice, the results are not highly sensitive to the precise value of \(\alpha\), as it mainly rescales the contribution of draws in the Laplacian weights.

Equilibrium Equations

Let:

\[ S_i = \sum_j \left(w_{ij} + \tfrac{1}{2} d_{ij}\right) \]

be the observed score of player \( i \), and

\[ E_i = \sum_j n_{ij} \left(P_{ij}^W + \tfrac{1}{2} P_{ij}^D\right) \]

the expected score under the model.

The equilibrium condition is:

\[ S_i = E_i, \quad \forall i \]

This corresponds to the first-order conditions of the likelihood system.

Linearization

We consider a reference point \(r^0\) and write:

\[ r_i = r_i^0 + \varepsilon_i \]

Moreover \( \varepsilon_i \) represents the tournament-induced correction. Let:

\[ E_i^{(0)} = E_i(r^0) \]

and define:

\[ F_{ij} = P_{ij}^W + \tfrac{1}{2} P_{ij}^D \]

A key property is that:

\[ \frac{\partial F_{ij}}{\partial \varepsilon_i} = p_{ij} q_{ij} + \frac{1}{4} t_{ij}(1 - t_{ij}) \]

\[ \frac{\partial F_{ij}}{\partial \varepsilon_j} = - \left( p_{ij} q_{ij} + \frac{1}{4} t_{ij}(1 - t_{ij}) \right) \]

This antisymmetry is essential and leads to a Laplacian structure.

The linearized equilibrium equations become:

\[ L \varepsilon = S - E^{(0)} \]

where the matrix elements are

\[ a_{ij} = n_{ij} \left( p_{ij} q_{ij} + \frac{1}{4} t_{ij}(1 - t_{ij}) \right) \]

The weights in the linear system coincide with the Fisher information of the underlying probabilistic model, meaning that each comparison is weighted according to its statistical informativeness.

\[ L_{ii} = \sum_j a_{ij}, \quad L_{ij} = -a_{ij} \]

Thus, \(L\) is a graph Laplacian.

The Laplacian system derived from the linearization is invariant under additive shifts of all player strengths, and is therefore singular.

To obtain a unique solution, we introduce a dummy opponent at the mean rating level. Each player is assigned a neutral interaction with this opponent, corresponding to a draw.

This construction fixes the reference level of the system and ensures that the resulting matrix is strictly positive definite.

Importantly, this regularization does not introduce any bias in relative rankings, as it only anchors the global reference level.

The ranking is obtained by ordering \(r_i^{post} = r_i^0 + \varepsilon_i\).

The system:

\[ L \varepsilon = S - E^{(0)} \]

can be interpreted as a global redistribution of discrepancies between observed and expected scores at the reference point \(r^0\).

This approximation is justified by two factors:

• the expansion point is already a strong prior, closely aligned with the true ranking in the simulated scenarios;
• the corrections introduced by the Laplacian system are typically small relative adjustments.

As a result, the linearized system captures the dominant structure of the problem while remaining computationally efficient.

In the limit:

\[ \alpha \to -\infty \]

draws disappear and the model reduces to the Zermelo model.

Relation to Iterative Methods

The Laplacian formulation can be interpreted as the first-order approximation of the nonlinear Davidson maximum likelihood problem, and is closely related to iterative optimization methods such as Newton or gradient-based updates.

To assess the accuracy of this approximation, we implemented the full nonlinear maximum likelihood estimation under the same model assumptions (Davidson with fixed parameters \(\alpha\) and \(\beta = 1/2)\). The nonlinear MLE consistently improves over the Laplacian solution across all ranking-sensitive metrics, with gains of approximately 10%.

This suggests that the Laplacian formulation already captures most of the information in the tournament graph, while nonlinear effects account for the remaining discrepancies.

Despite this improvement, the Laplacian formulation remains central for several reasons. First, it reveals the intrinsic structure of the problem: the ranking emerges as the solution of a linear system on the comparison graph, with weights related to local Fisher information. This provides a direct interpretation of global consistency across pairwise comparisons.

Second, the resulting system is linear, symmetric, and well-posed, ensuring a unique and stable solution without the need for iterative optimization. Third, it establishes direct connections with spectral methods and graph-based ranking approaches.

In this sense, the Laplacian model should be viewed as the structural backbone of the method, while the nonlinear MLE represents a refinement layer.

Choice of the Reference Point

The method depends on the choice of the expansion point \(r^0\), but not structurally. Different choices lead to different approximations:

• \(r^0 = 0\): purely tournament-based model,
• arbitrary \(r^0\): general linearization,
• iterative updates of \(r^0\): recovery of Newton-type methods.

A natural and effective choice for \(r^0\) is provided by the Elo rating system, due to:

• its empirical reliability,
• its consistency with logistic models,
• its ability to provide a meaningful global baseline.

So we use the Elo ratings \( R_i \):

\[ r_i^0 = \frac{R_i - \bar{R}}{s}, \quad s = \frac{400}{\ln 10} \]

Experimental Methodology

To evaluate the proposed methods, we adopt a controlled simulation framework in which a ground-truth ranking is available. This allows us to measure not only correlation with tournament standings, but also the ability of each method to recover the underlying ordering.

The comparison includes the following methods:

• Laplacian linearization of the Davidson model, LAP_Davidson,
• Laplacian linearization of the Zermelo model, LAP_Zermelo,
• Official Swiss standings with standard tie-breaks, OFF_1, OFF_2,
• A baseline ranking based solely on pre-tournament ratings, EloOnly.

Ground Truth Construction

A complete round-robin tournament is first generated for N players.

• Each player is assigned a latent strength parameter, i.e. an Elo rating.
• Match outcomes are generated probabilistically according to a model consistent with FIDE paired comparisons (Milvang, 2016).
• The full round-robin results are used to compute a reference ranking, which serves as the ground truth.

This step ensures that the “true ordering” is based on a dense comparison graph, eliminating pairing artifacts.

Swiss Pairing Simulation

Swiss tournaments are generated from the same pool of players used in the round-robin ground truth. The pairing procedure is designed to mimic the structural properties of real Swiss systems while remaining fully controlled.

At each round, players are first grouped by their current score (score groups). Within each score group we can choose if players are:

• randomly permuted, or
• sorted by rating.

We simulated both scenarios because the outcome may induce in the underlying graph unexpected structure that may affect the final ranking.

Pairings are then determined through a global weighted matching procedure, with the following principles:

• pairings within the same score group are preferred,
• in each scoregroup pairings are performed with usual rule #1 vs #(n/2+1), etc., where \(n\) is the number of players in each score group
• colors balancing and floater limitations are not considered.

Although the pairings are not identical to current FIDE rules, these rules introduce systematic operations during the pairings that, if present, should affect the graph connectivity.

Match outcomes are not simulated independently. Instead, they are retrieved from the pre-generated round-robin tournament:

• for each paired pair \((i,j)\), the result is taken directly from the round-robin data,
• thus, all matches are consistent with a single underlying probabilistic realization.

This design has two key advantages:

• 1. Realistic pairing structure

The Swiss tournaments reflect the combinatorial constraints of real competitions:

• sparse comparison graph,
• clustering by score groups,

• limited cross-group interactions.

• 2. Controlled outcome model

By reusing round-robin results:

• all comparisons are embedded in a single global structure,
• the “true ranking” is well-defined,
• differences between methods arise only from ranking algorithms, not from additional randomness.

The experiment isolates the effect of pairing-induced sparsity and score-group constraints, while keeping the underlying performance structure fixed.

Choice of Evaluation Metrics

In our setting, a ground-truth ranking is available, obtained from a full round-robin tournament. The goal is therefore not to define an abstract notion of ranking quality, but rather to assess how well different methods are able to reproduce this reference ordering.

In particular, we are interested in identifying which ranking procedures best approximate the true final ranking when only partial information, as in a Swiss tournament, is available.

From a practical perspective, our primary concern is not a generic correlation measure, but the accuracy of the ranking in positions that matter, namely those associated with prizes and titles.

While it is unrealistic to expect that a small number of Swiss rounds can accurately recover the full ordering of N players, it is reasonable to require that a good method should correctly identify the top positions.

In typical tournaments, prizes are concentrated among the leading players. For this reason, we focus in particular on the ability of each method to reproduce the top 10 positions of the ground-truth ranking.

More generally, since the Swiss system can be interpreted as a sparse sampling of an underlying round-robin tournament, an effective ranking method should be able to learn from the available data and approximate the ranking that would emerge from the full set of comparisons.

To assess the robustness of the ranking methods under different assumptions on player performance, we consider three distinct models for generating match outcomes in the underlying round-robin tournament.

These scenarios allow us to evaluate how methods behave under:

• idealized conditions,
• realistic variability,
• and complete absence of structure.

(a) FIDE Mode (Deterministic Rating Model)

In this scenario, player ratings are assumed to be fixed and stable across all games. Match outcomes are generated according to the probabilistic model proposed by the FIDE Technical Commission in their statistical framework for tournament simulations (Milvang, 2016).

The probabilities of win, draw, and loss depend solely on the rating difference between players. This represents an idealized setting, where:

• player strength is constant,
• all randomness is captured by the probabilistic model,
• no external variability (form, fatigue, etc.) is present.

(b) REAL Mode (Noisy Performance Model)

In this scenario, player performance fluctuates from game to game.

Before each match, ratings are perturbed independently by Gaussian noise:

\[ r_i^{\text{eff}} = r_i + \mathcal{N}(0, \sigma) \]

The outcome is then sampled using these perturbed ratings. This models day-to-day variability in player performance, including:

• form fluctuations,
• psychological factors,
• external conditions.

Typical values of \(\sigma\) (e.g., 50–100 Elo points) produce realistic upset frequencies (e.g., a significant chance of reversal in a 200-point gap).

This scenario is intended to approximate real tournament conditions.

(c) RANDOM Mode (No Skill Structure)

In this baseline scenario, match outcomes are generated independently of player ratings. Each result (win, draw, loss) occurs with equal probability:

\[ P(\text{win}) = P(\text{draw}) = P(\text{loss}) = \frac{1}{3} \]

This represents a null model, where:

• no underlying skill structure exists,
• rankings cannot be meaningfully recovered from data.

These three regimes serve complementary purposes:

• FIDE mode evaluates performance under ideal statistical assumptions,
• REAL mode tests robustness under realistic noise,
• RANDOM mode verifies that methods do not extract spurious structure from pure randomness.

A natural choice in ranking evaluation is the use of rank correlation indices, such as Spearman’s coefficient. However, in the present context, such measures turn out to be largely uninformative.

In our simulations, we observe that a ranking based solely on pre-tournament ratings (i.e., without using any game results) already achieves extremely high correlation with the final round-robin ranking:

• REAL mode: Spearman ≈ 0.989
• FIDE mode: Spearman ≈ 0.991

Despite these very high values, a closer inspection reveals that the top positions differ substantially, with only partial overlap among the top 10 players.

This highlights a fundamental limitation: High global rank correlation does not imply accuracy where it matters most.

The issue arises from the structure of the ranking problem.

• Spearman correlation is dominated by the bulk of the ranking.
• Small local discrepancies at the top positions contribute negligibly to the overall index.
• As a result, two rankings can have almost perfect correlation, yet significantly different top-10 sets.

Therefore a ranking method that correctly identifies the top players but differs elsewhere is preferable to one that achieves high global correlation but misidentifies the leaders.

In our setting, Spearman correlation is essentially saturated by the prior rating structure and fails to discriminate between methods that differ significantly in their ability to identify the top-ranked players.

For these reasons, we do not rely on Spearman correlation as a primary evaluation metric, although we report it. Instead, we focus on measures that directly assess:

• accuracy in top positions,
• recovery of critical pairwise inversions,
• ability to extract meaningful structure from sparse data.

Top-(k) Accuracy

We first consider the overlap between the estimated ranking and the ground-truth ranking restricted to the top positions.

• Top-5 overlap
• Top-10 overlap

These metrics count how many players in the top (k) of the true ranking also appear in the top (k) of the estimated ranking.

They measure the ability to correctly identify prize-winning players.

Unlike Spearman, these metrics are directly aligned with tournament objectives, where only a small number of positions determine prizes and titles.

Cross-Group Kendall Tau (CGTau)

Swiss tournaments impose a strong structural constraint: players are primarily ranked by score, and tie-breaks only act within score groups.

To evaluate the ability of a method to go beyond this limitation, we define:

\[ \text{CGTau} \]

as a Kendall-type correlation computed only on pairs of players belonging to different score groups.

It measures how well a ranking orders players across score group boundaries and ignores trivial within-group reorderings.

This isolates the hard part of the problem, namely comparing players who did not face each other directly.

Inversion Recovery (InvRec)

A central issue in Swiss tournaments is the presence of cross-scoregroup inversions. These occur when:

• player (i) has more Swiss points than player (j),
• but in the ground truth (j) is actually stronger (higher in ranking).

We define Inversion Recovery (InvRec) as:

\[ \text{InvRec} = \frac{\text{number of correctly recovered inversions}} {\text{total number of true inversions}} \]

A key property is that Official rankings always have InvRec = 0 because they are strictly ordered by score.

Thus this index measures the ability of a ranking method to correct structural errors introduced by the Swiss pairing system.

To emphasize the importance of high-ranking players, we also consider a restricted version:

Inv@Top10

This metric counts only inversions involving players belonging to the top fraction of the ground-truth ranking.

It evaluates whether the method correctly identifies strong players who are underestimated by the Swiss score.

This is particularly relevant for:

• prize allocation,
• title norms,
• qualification thresholds.

Conditioned Rank Error

We further analyze the effect of ranking adjustments by distinguishing between:

• players whose position remains within their score group,
• players whose position is moved outside their score group.

We define:

• displaced rank error: average error for players moved across groups,
• non-displaced rank error: average error for players left within their group.

The logic is that a useful method should improve accuracy precisely where it modifies the ranking.

This metric allows us to verify whether cross-group corrections are:

• meaningful (error decreases), or
• noisy (error increases).

The proposed metrics are not generic ranking measures, but are tailored to the specific structure of Swiss tournaments and to the practical objectives of competitive play.

All metrics are averaged over 100 simulations, each one consisting of a Round Robin and Swiss tournament.

This reduces variance and provides stable estimates of performance.

Rankings Compared

For each simulated tournament, the following rankings are computed included the official ones.

It is important to emphasize that, beyond the primary ordering by score, there is no universally prescribed method for producing a final ranking in chess tournaments.

While governing bodies such as FIDE provide a list of commonly used tie-break systems, the choice and ordering of these tie-breaks are not uniquely fixed.

In practice:

• organizers and federations select tie-break rules,
• and their order of application,
• based largely on tradition and empirical usage, rather than on a formally established optimality criterion.

As a consequence different choices of tie-break ordering can produce different final rankings, even when based on the same tournament results. This reflects a broader issue: there is no clear theoretical consensus on the relative validity of the available tie-break methods. While the issue remains open, new tie-break criteria are periodically introduced, reflecting the lack of a universally accepted solution.

In this work, we consider two representative official rankings for swiss tournament:

• OFF_1: score → Buchholz → Buchholz Cut-1

• OFF_2: score → Buchholz Cut-1 → Buchholz

These choices reflect commonly used variants in practice and allow us to assess the sensitivity of results to the ordering of tie-break criteria.

Despite their differences, both OFF_1 and OFF_2 share a key structural property:

• players are strictly ordered by score groups,
• tie-breaks only operate within each group,
• no cross-group correction is possible.

As a consequence:

• InvRec = 0 by construction,
• ranking differences between OFF_1 and OFF_2 are limited to within-group reorderings.

The official rankings serve as:

• a practical benchmark, reflecting current tournament procedures,
• a reference point for comparison with model-based methods,
• an illustration of the inherent limitations of score-based ranking systems.

The others rankings used for comparison are:

• LAP_Davidson: linearization of the Davidson model (with draws), i.e. this work outlined above,
• LAP_Zermelo: linearization of the Zermelo model (no draws).

Both methods:

• solve a single global linear system,
• use a minimal dummy regularization (one pseudo-opponent per player),

• differ only in the underlying probabilistic model and resulting weights.

• EloOnly: ranking induced by pre-tournament ratings, ignoring tournament results.

Results

All results are averaged over 1000 simulated tournaments with 100 players (ratings uniformly distributed in the range 1600–2400). Each tournament consists of 9 Swiss rounds.

We consider three outcome-generation regimes:

• RANDOM: no underlying skill structure,
• FIDE: deterministic rating-based performance,
• REAL: noisy performance with game-by-game fluctuations.

In addition, we distinguish between:

• deterministic pairing (FIDE-like),
• randomized scoregroups (neutralized pairing).

The results are reported using the metrics introduced in Section 10, with particular emphasis on Top-k accuracy, CGTau, and Inversion Recovery.

Table 1 — RANDOM Mode (No Skill Structure)

In this regime, match outcomes are purely random, so no latent ranking structure exists. All methods perform poorly, as expected. In particular, EloOnly collapses to near-zero correlation, confirming that the rating prior contains no useful information in this setting. Official rankings slightly outperform the model-based methods in RankErr and Top-k metrics, simply because they exploit score aggregation. By contrast, the Laplacian methods do not extract strong spurious structure from noise, which is a desirable property. This table therefore acts as a sanity check: when no signal is present, no method should appear artificially strong.

Table 2 — FIDE Mode (Deterministic Rating-Based Pairing)

This is the most important table. The Swiss tournament is now generated with a deterministic pairing scheme closer to the FIDE logic, and the main phenomenon becomes even clearer. EloOnly remains extremely strong, showing that the true ranking is still closely aligned with the initial ratings. However, the official rankings degrade dramatically: RankErr rises above 22, Top-10 recovery drops to nearly 3 out of 10, and InvRec remains zero by construction. The gap between OFF and LAP_Davidson is now substantial across all meaningful metrics. This strongly supports the central claim of the paper:

deterministic Swiss pairing induces structural bias in the comparison graph, and score-based rankings are unable to correct it.

LAP_Davidson substantially improves RankErr, CGTau, InvRec, and Top-10 overlap, while LAP_Zermelo also improves over OFF but remains clearly weaker than the Davidson version. This confirms that draw information contributes meaningfully to recovering latent ranking structure.

Table 3 — FIDE Mode with Randomized Scoregroups

When scoregroups are randomized, the graph becomes more mixed and less structurally constrained. Official rankings recover dramatically compared with the deterministic case: RankErr is reduced by more than half, and Top-10 overlap rises from about 3.15 to about 5.84. The gap between OFF and LAP methods becomes much smaller. This confirms the key interpretation already suggested by the previous experiments:

the degradation is not intrinsic to the Swiss system itself, but is largely induced by deterministic pairing rules.

LAP_Davidson still performs slightly better overall, especially in CGTau and inversion recovery, but the advantage is now much less pronounced. This is exactly what one would expect if the graph distortion has been reduced at the source.

Table 4 — REAL Mode (Noisy Performance, Deterministic Pairing)

This scenario adds realistic game-to-game fluctuations, while keeping deterministic rating-based pairing. The main conclusions remain unchanged and are in fact reinforced. Official rankings perform very poorly, with RankErr above 23 and Top-10 overlap close to 3. By contrast, LAP_Davidson continues to produce a clear improvement in all important metrics: lower rank error, higher CGTau, and substantial inversion recovery. LAP_Zermelo remains weaker, especially in the treatment of cross-group inversions. This shows that the advantage of the Davidson-based Laplacian approach is robust under realistic noise, not just under an idealized fixed-rating model.

Across all structured scenarios, a stable pattern emerges:

1. In the absence of structure (RANDOM), no method performs well, as expected.
2. Under deterministic pairing, official rankings degrade severely.
3. Under randomized scoregroups, official rankings recover strongly.
4. LAP_Davidson consistently improves over the official rankings in the deterministic cases.
5. LAP_Davidson also consistently outperforms LAP_Zermelo, showing that draw information helps recover latent structure.

Impact of Pairing Rules

The results presented in the previous section reveal a fundamental structural property of Swiss-system tournaments:

the pairing mechanism is not neutral with respect to ranking accuracy.

This aspect is often overlooked, as Swiss systems are typically viewed as procedures that simply generate a sparse subset of pairwise comparisons. However, the way in which pairings are constructed has a direct impact on the structure of the comparison graph, and consequently on the quality of the inferred ranking.

A Swiss tournament can be interpreted as a process that builds a comparison graph over the set of players, where edges correspond to played games.

Under this perspective, different pairing rules induce different graph topologies:

• Randomized pairing within scoregroups produces a relatively homogeneous and well-mixed graph;
• Deterministic rating-based pairing, as in current FIDE practice, produces a structured and highly clustered graph.

In the deterministic case, players with similar ratings tend to be paired repeatedly. As a result:

• strong players mainly face strong opponents,
• weaker players mostly play among themselves,
• interactions across different strength levels are limited.

This leads to a graph that is locally dense but globally poorly connected, with reduced information flow between distant regions of the ranking.

Score-based rankings, including official systems based on tie-breaks such as Buchholz, implicitly rely on the assumption that the comparison graph provides a sufficiently representative sampling of player interactions.

When the graph is highly structured, this assumption is violated. In particular:

• tie-break values depend on the strength of opponents,
  but opponent strength is itself affected by the same pairing constraints;
• limited cross-group interactions prevent reliable comparisons between players with different scores;
• as a consequence, structural biases are introduced in the ranking.

This phenomenon is clearly visible in the FIDE pairing simulations, where official rankings exhibit a significant degradation in all accuracy metrics, despite the strong predictive power of the initial ratings.

When scoregroups are randomized, the pairing becomes effectively less structured. Empirically, we observe that:

• the comparison graph becomes more uniformly connected,
• cross-level interactions increase,
• score-based rankings improve significantly.

This demonstrates that:

the degradation observed under deterministic pairing is not intrinsic to the Swiss system itself, but is a consequence of the specific pairing rules.

The proposed ranking method operates on the entire comparison graph and solves a global consistency problem.

Unlike score-based rankings, it:

• aggregates information along indirect paths,
• is not constrained by score groups,
• propagates information across different regions of the graph.

As a result, it can partially compensate for the lack of direct comparisons between players.

This explains the consistent improvement observed in:

• cross-group ordering (CGTau),
• inversion recovery (InvRec),
• and top-position accuracy.

Recent work by Sauer et al. (2021) has shown that modifying the pairing mechanism can significantly improve ranking accuracy in Swiss tournaments.

Their approach replaces the rule-based FIDE pairing system with an optimization-based matching procedure, resulting in better mixing of players and improved ranking quality.

Our findings provide a structural explanation for this phenomenon: deterministic pairing rules induce biases in the comparison graph, and reducing these constraints leads to more informative tournaments.

In contrast to their approach, we do not modify the pairing mechanism. Instead, we show that:

it is possible to correct the distortions introduced by standard pairing rules at the ranking level.

The results highlight an inherent trade-off in the design of Swiss tournaments:

deterministic pairing improves: reproducibility and transparency, but reduces statistical efficiency and ranking accuracy.

These findings suggest two complementary directions for improving Swiss tournaments:

1. pairing-level improvements, by designing pairing rules that enhance graph connectivity (as in Sauer et al.);
2. ranking-level corrections, by applying global statistical methods that compensate for structural biases.

The approach proposed in this paper belongs to the second category. It should therefore be interpreted not as a replacement for Swiss tournaments, but as:

a correction layer that improves ranking accuracy while preserving the existing pairing structure.

Comparison with Least Squares Methods

As discussed in previous work (Forlano, 2026), we compare the proposed linearized Davidson approach with both the bidimensional Zermelo model and the least-squares ranking method introduced by Csató (2017).

The following table reports the rankings obtained for the European Team Championship under different methods:

A notable feature of the table is the close agreement between the LAP_Davidson and least-squares (LS) rankings, particularly in the upper part of the table where the most relevant positions are concentrated.

This observation suggests that the two approaches may be capturing similar structural information from the comparison graph.

From a mathematical perspective, this is not entirely surprising. Least-squares ranking methods are known to lead to Laplacian linear systems, obtained by minimizing squared discrepancies of pairwise differences. The present approach also results in a Laplacian system, derived from a first-order linearization of a probabilistic model.

Despite this structural similarity, the two methods originate from different principles:

• the least-squares approach is based on an algebraic fitting criterion;
• the present method is derived from a likelihood-based model that explicitly incorporates draw probabilities.

The empirical proximity of the resulting rankings therefore raises an interesting question:

to what extent do different Laplacian-based approaches encode the same underlying ranking information?

A detailed investigation of this relationship is beyond the scope of the present work and is left for future research.

Finally, we observe that all methods produce nearly identical rankings in the lower part of the table, while differences are concentrated in the upper and middle regions. This suggests that the relevant distinctions between ranking methods emerge precisely where the comparison graph is most informative.

Conclusion

The main conclusion of this work is that Swiss tournaments should not be interpreted as neutral sampling procedures.

Deterministic pairing rules, such as those used in current FIDE systems, introduce structural biases in the comparison graph, leading to a significant degradation of score-based rankings.

This explains why alternative pairing mechanisms, such as those proposed by Sauer et al. (2021), improve ranking accuracy.

Instead of modifying the pairing system, we propose a complementary approach: correcting the induced distortions at the ranking level through a graph-based method.

This perspective opens the way to hybrid systems in which traditional Swiss tournaments are combined with statistically grounded ranking corrections.

More generally, these results show that ranking accuracy in Swiss tournaments cannot be assessed independently of the pairing mechanism that generates the comparison graph.

References

• Csató, L. (2017). Ranking in Swiss system chess team tournaments. Annals of Operations Research, 254(1–2), 17–36. https://doi.org/10.1007/s10479-017-2440-4

• Luigi Forlano, 2026, A Statistical Model for Chess Tournaments with Draws.

• Otto Milvang, 2016, Probability for the outcome of a chess game based on rating, https://pairings.fide.com/images/stories/downloads/2016-probability-of-the-outcome.pdf.

• Pascal Sauer, Ágnes Cseh, Pascal Lenzner, 2023, Improving Ranking Quality and Fairness in Swiss-System Chess Tournaments, https://arxiv.org/pdf/2112.10522

version 1 : 07 April 2026