NOTE TO READERS Here is a revisitation of my earlier “Missing Point Score System”, now developed on a more solid theoretical basis. I present a statistical model for chess tournaments with draws, based on a bidimensional latent representation of players (strength and draw propensity), which leads to a global ranking independent of the pairing structure. However, I am not in a position to fully assess whether this approach is genuinely original or how it relates to existing lines of research that I may not be aware of. If similar ideas have already been developed, I would be very grateful for feedback helping to identify the relevant references and to properly acknowledge prior contributions. Any comments or suggestions are warmly appreciated. 21 March 2026

A Statistical Model for Chess Tournaments with Draws: A Bidimensional Approach

Abstract

We propose a statistical model for chess tournaments that explicitly accounts for draws while preserving the structural properties of classical paired-comparison models.

In the traditional Zermelo and Bradley–Terry frameworks, match outcomes depend only on relative strength, making them unable to capture the empirically observed increase in draw frequency among stronger players. Extensions such as Davidson’s model introduce a global draw parameter, but do not reflect player-specific drawing tendencies or strength-level effects.

We introduce a two-dimensional latent structure in which each player is characterized by a competitive strength and a solidity parameter governing draw propensity. Match outcomes are modeled through a simple probabilistic mechanism in which wins depend on strength, while draws emerge from the interaction of player solidities.

The resulting model retains global concavity of the log-likelihood, ensuring stable maximum likelihood estimation. We derive explicit expressions for the gradient and the information matrix, leading to an efficient Newton method.

The model induces a natural ranking based on expected scores in a virtual round-robin against the entire field, thereby decoupling ranking from the pairing process.

This approach provides a coherent and tractable framework for analyzing chess tournaments with draws.

Introduction

Statistical models for paired comparisons have a long tradition, dating back to the seminal work of Zermelo (1929), who proposed a probabilistic framework for estimating player strengths from match results. This model, later rediscovered and generalized as the Bradley–Terry model, assumes that the probability of a win depends only on the ratio of the players’ strengths.

However, in many real-world applications—most notably chess—matches can end in a draw. This feature poses a fundamental challenge for classical models, which are inherently binary.

A widely used extension is the model of Davidson (1970), which introduces a draw parameter to account for tied outcomes. While mathematically convenient, this approach assumes a constant draw tendency across all players, thus failing to capture the well-documented empirical fact that draws become more frequent among stronger players.

Several attempts have been made to incorporate strength-dependent draw probabilities by introducing additional global parameters or functional corrections. However, such modifications typically lead to a loss of concavity in the log-likelihood function, resulting in unstable estimation procedures and multiple local optima.

In parallel, alternative ranking methods have been proposed for chess tournaments, including least-squares approaches such as those studied by Csató (2017). These methods provide robust rankings from incomplete comparison structures, although they are not derived from an explicit generative model of match outcomes.

Empirical studies and practical applications show that such methods often produce rankings that differ systematically from official standings based on tie-break criteria. This indicates that the discrepancies are not specific to a particular ranking technique, but rather reflect structural features of the Swiss system itself.

From this perspective, the objective of the present work is not to compare alternative ranking methods. Instead, we focus on a more fundamental issue: the path dependence and limited informational content of Swiss-system tournaments. The proposed model should therefore be understood as a representative example of a global inference approach, rather than as a competing ranking procedure.

We propose a bidimensional latent representation of players, in which each participant is characterized by both a competitive strength and a draw propensity parameter. This formulation allows draw probabilities to emerge naturally from player interactions, without introducing global parameters and while preserving the concavity of the likelihood function.

The resulting model retains the desirable properties of classical paired comparison frameworks while providing a more realistic and structurally consistent treatment of draws.

Motivation

In the classical model of Ernst Zermelo (1929), the probability that player \(i\) defeats player \(j\) is

\[ P_{ij} = \frac{u_i}{u_i + u_j}, \]

where \(u_i > 0\) represents the strength of player \(i\).

A key limitation of this formulation is that probabilities depend only on the ratio of strengths, not on their absolute level.

As a consequence:

• two very strong players have the same draw probability as two weak players with the same rating difference;
• in particular, draws are not modeled at all.

The extension proposed by Davidson (1970) introduces a global draw parameter \(\eta\), but:

• \(\eta\) is constant across all players;
• it does not capture the well-known empirical fact that draws become more frequent among stronger players.

A natural idea is to make the draw parameter depend on the absolute strength level, for example

\[ \eta_{ij} = \eta \cdot f(u_i, u_j). \]

However, such approaches lead to a fundamental issue:

the log-likelihood typically loses concavity

This has severe consequences:

• unstable optimization;
• possible multiple local maxima;
• loss of the clean structure of Zermelo / Bradley–Terry models.

Attempts to incorporate draws using a single global parameter, or even level-dependent global functions, are unsatisfactory if one wants to preserve convexity and robustness.

A Two-Dimensional Model

We assign to each player two independent parameters:

• \(r_i\): log-strength (ability to win);
• \(\sigma_i\): log-solidity (propensity to draw).

Define

\[ u_i = e^{r_i}, \qquad s_i = e^{\sigma_i}. \]

These quantities admit a natural interpretation:

• \(u_i\): strength (offensive potential);
• \(s_i\): solidity (draw propensity).

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

\[ Z_{ij} = u_i + u_j + s_i s_j. \]

Then:

\[ P_{ij}^{W} = \frac{u_i}{Z_{ij}}, \qquad P_{ij}^{L} = \frac{u_j}{Z_{ij}}, \qquad P_{ij}^{D} = \frac{s_i s_j}{Z_{ij}}. \]

The probability of a draw satisfies

\[ P_{ij}^{D} \propto s_i s_j, \]

that is, it is proportional to the product of the players’ solidities. Thus, draws emerge from the interaction of player solidities, not from an external parameter.

If \(s_i \to 0\) for all players,

\[ P_{ij}^{W} \to \frac{u_i}{u_i + u_j}, \]

and the model reduces exactly to Zermelo’s model. Thus, the classical formulation is recovered as a special case.

The model is characterized by:

• no exogenous draw parameter;
• no level-dependent correction;
• all parameters are player-specific.

Log-Likelihood

For aggregated match data \((w_{ij}, d_{ij}, l_{ij})\), define

\[ n_{ij} = w_{ij} + d_{ij} + l_{ij}. \]

The contribution of pair \((i,j)\) is

\[ \ell_{ij} = w_{ij} r_i + l_{ij} r_j + d_{ij} (\sigma_i + \sigma_j)- n_{ij} \log Z_{ij}, \]

with

\[ Z_{ij} = e^{r_i} + e^{r_j} + e^{\sigma_i + \sigma_j}. \]

The total log-likelihood is

\[ \ell(r,\sigma) = \sum_{i<j} \ell_{ij}. \]

For each pair \((i,j)\), the contribution \(\ell_{ij}\) is the sum of a linear term and the negative of a log-sum-exp function of affine forms. Since log-sum-exp is convex, each \(\ell_{ij}\) is concave in \((r,\sigma)\). Therefore, the total log-likelihood \(\ell (r,\sigma)\), being a sum of concave terms, is globally concave.

Gradient

For each pair \((i,j)\):

\[ \frac{\partial \ell_{ij}}{\partial r_i} = w_{ij} - n_{ij} P_{ij}^{W}, \qquad \frac{\partial \ell_{ij}}{\partial r_j} = l_{ij} - n_{ij} P_{ij}^{L}, \]

\[ \frac{\partial \ell_{ij}}{\partial \sigma_i} = d_{ij} - n_{ij} P_{ij}^{D}, \qquad \frac{\partial \ell_{ij}}{\partial \sigma_j} = d_{ij} - n_{ij} P_{ij}^{D}. \]

Thus the gradient has the natural form

\[ \text{observed} - \text{expected}. \]

The model has one degree of freedom corresponding to a joint shift of r and σ:

\[ r_i \to r_i + c, \qquad \sigma_i \to \sigma_i + \frac{c}{2}. \]

A normalization such as

\[ \sum_i r_i = 0. \]

is sufficient to fix the scale. This invariance reflects a structural coupling between strength and solidity, implying that only relative differences are identifiable without normalization.

In this model each player corresponds to a point in

\[ \mathbb{R}^2, \]

with:

• first coordinate: strength;
• second coordinate: solidity.

The model defines a two-dimensional embedding of players.

Ranking

Since strength is now two-dimensional, no natural total order exists.

We define a Virtual Round Robin score:

\[ E_{ij} = P_{ij}^{W} + \frac{1}{2} P_{ij}^{D}, \]

\[ S_i = \sum_{j \ne i} E_{ij}. \]

This corresponds to the classical scoring system \((1, 1/2, 0)\) used in chess tournaments. Alternative scoring systems could be incorporated without modifying the probabilistic model.

The ranking is obtained by ordering \(S_i\). We refer to \(S_i\) as Zermelo score and to the induced ranking as Zermelo ranking.

\[ S_i = \text{expected score against the entire field}. \]

This ranking is:

• global;
• model-consistent;
• independent of the actual pairing system.

Existence of the MLE

A subtle but important issue arises. If a player \(i\) has no draws at all, then

\[ \frac{\partial \ell}{\partial \sigma_i} < 0, \]

which implies that the likelihood increases as

\[ \sigma_i \to -\infty. \]

Hence the MLE may fail to exist in the interior of the parameter space, and boundary solutions may occur.

To ensure numerical stability, we consider

\[ \ell_\lambda(r,\sigma) = \ell(r,\sigma)- \frac{\lambda_r}{2}\sum_i r_i^2- \frac{\lambda_\sigma}{2}\sum_i \sigma_i^2. \]

This:

• prevents divergence;
• improves conditioning;
• preserves the structure of the model.

Maximum Likelihood Estimation

We now describe the estimation procedure in detail. Let

\[ x = (r_1,\dots,r_N,\sigma_1,\dots,\sigma_N)^T \in \mathbb{R}^{2N}. \]

We maximize the penalized log-likelihood

\[ \ell_\lambda(r,\sigma) = \ell(r,\sigma)- \frac{\lambda_r}{2}\sum_i r_i^2- \frac{\lambda_\sigma}{2}\sum_i \sigma_i^2. \]

For each unordered pair \((i,j)\), define

\[ a = e^{r_i}, \quad b = e^{r_j}, \quad c = e^{\sigma_i+\sigma_j}, \quad Z = a + b + c. \]

\[ p = \frac{a}{Z}, \quad q = \frac{b}{Z}, \quad t = \frac{c}{Z}. \]

The log-likelihood contribution is

\[ \ell_{ij} = w_{ij} r_i + l_{ij} r_j + d_{ij}(\sigma_i+\sigma_j)- n_{ij}\log Z. \]

For each pair \((i,j)\), the contributions to the gradient are

\[ g_{r_i} \mathrel{+}= w_{ij} - n_{ij} p, \]

\[ g_{r_j} \mathrel{+}= l_{ij} - n_{ij} q, \]

\[ g_{\sigma_i} \mathrel{+}= d_{ij} - n_{ij} t, \]

\[ g_{\sigma_j} \mathrel{+}= d_{ij} - n_{ij} t. \]

With penalization:

\[ g_{r_i} \mathrel{-}= \lambda_r r_i, \qquad g_{\sigma_i} \mathrel{-}= \lambda_\sigma \sigma_i. \]

We work with the information matrix

\[ I = -\nabla^2 \ell_\lambda, \]

which is positive semidefinite (and positive definite after regularization).

For each pair \((i,j)\), the contribution to \(I\), in the variables

\[ (r_i, r_j, \sigma_i, \sigma_j), \]

is

This structure is particularly convenient, since each pair updates only a small block of the global matrix. The per-pair block is rank-deficient, but the global information matrix becomes full rank after aggregation and regularization.

Regularization adds:

\[ I_{r_i,r_i} \mathrel{+}= \lambda_r, \qquad I_{\sigma_i,\sigma_i} \mathrel{+}= \lambda_\sigma. \]

Because of the identifiability constraint

\[ \sum_i r_i = 0, \]

we solve a constrained Newton system.

Let

\[ C = (1,\dots,1,0,\dots,0) \in \mathbb{R}^{1 \times 2N}. \]

At each iteration, we solve the KKT system

We then update

\[ x_{\text{new}} = x + \alpha \, \delta x, \]

where \(0 < \alpha \le 1\) is chosen by a line search.

Each iteration of the algorithm consists of:

1. initialize \(\ell, g, I = 0\);
2. loop over all observed pairs \((i,j)\);
3. compute \(p, q, t\);
4. accumulate likelihood, gradient, and information matrix;
5. add regularization;
6. solve the KKT system;
7. perform line search;
8. update parameters.

A practical initialization is:

\[ r_i = 0 \quad \forall i, \]

and a common value for \(\sigma_i\), derived from the global draw rate \(\bar d\):

\[ \sigma^{(0)} = \frac{1}{2} \log\left(\frac{2\bar d}{1-\bar d}\right). \]

This corresponds to the homogeneous case where all players are identical.

Computational remarks:

• gradient and Hessian assembly cost is \(O(m)\), where \(m\) is the number of observed pairs;
• the linear solve is \(O((2N)^3)\) in dense form;
• for typical tournament sizes, this is computationally feasible;
• the block structure allows efficient sparse implementations.

Global concavity ensures robust convergence of Newton's method.

Comparison and Discussion

Case 1: European Team Championship 2011

The model was fitted to the 18th European Team Chess Championship 2011 Open, a 38-team Swiss-system event played over 9 rounds. This tournament is also one of the case studies analysed by Csató, which makes it a natural benchmark for comparison. The official final standings were headed by Germany, followed by Azerbaijan and Hungary (https://chess-results.com/tnr57856.aspx).

A typical reduced output of the program is the following.

The estimation converges rapidly: the penalized log-likelihood improves monotonically from \(-179.54\) to \(-94.09\), and the gradient norm falls from about \(10.96\) to less than \(8 \times 10^{-4}\) in only 10 iterations, indicating that the Newton scheme is numerically stable and effective on a real Swiss tournament. The full output is available from the author upon request.

The optimization history shows:

• convergence in 10 iterations;
• final log-likelihood (-94.09469);
• very small terminal gradient norm;
• excellent calibration of observed and expected scores.

In particular, the reported fit statistics are:

\[ \text{MAE}=0.006,\qquad \text{RMSE}=0.008,\qquad \text{Bias}\approx 0,\qquad \text{MaxAbsErr}=0.028. \]

These values indicate that the fitted model reproduces the observed tournament scores very closely.

The model ranks the top teams as follows:

1. Germany
2. Azerbaijan
3. Bulgaria
4. Russia
5. Armenia
6. Hungary
7. Romania
8. Spain
9. Netherlands
10. Italy.

In the official final table, the top five were Germany, Azerbaijan, Hungary, Armenia, and Russia, with Bulgaria in seventh place. By contrast, our model promotes Bulgaria to third place and places Hungary sixth. This difference is a direct consequence of using a network-based inference model rather than a lexicographic ranking driven by the short comparison path induced by the pairing algorithm

The estimated parameters are especially informative because they separate two different aspects of performance:

• \(r_i\): the team’s ability to score wins against the field;
• \(\sigma_i\): the team’s tendency to generate draws.

For instance, Germany and Azerbaijan have very similar estimated strength parameters \(r\), but Azerbaijan has a substantially larger \(\sigma\), reflecting its more draw-prone profile. Germany finished with 7 wins, 1 draw, and 1 loss, while Azerbaijan scored 6 wins, 2 draws, and 1 loss.

Likewise, Hungary has a relatively high solidity parameter, consistent with its record of 5 wins, 3 draws, and 1 loss, whereas Bulgaria combines a strong \(r\) with a lower \(\sigma\), reflecting a more decisive performance profile: 5 wins, 2 draws, and 2 losses. In other words, Hungary appears more “solid”, while Bulgaria appears more “aggressive” or “decisive”. This shows how teams with similar final scores may have different structural profiles that are not captured by traditional standings.

This is one of the main advantages of the bidimensional approach: teams with similar final scores can be distinguished structurally. A one-dimensional model must compress everything into a single number; here, instead, the model can represent the difference between a team that often wins and loses, and a team that draws more frequently against strong opposition.

Another notable feature of the output is the near-perfect agreement between observed score and expected score for each team. For the top teams, the discrepancy is about \(0.01\) points or less, and even at the bottom of the table the absolute error remains very small.

The algorithm also outputs standard errors and \(t\)-ratios for both \(r_i\) and \(\sigma_i\). For the leading teams, the strength coordinate \(r_i\) is clearly estimated with strong signal, while for some lower teams the solidity estimates are much more uncertain. This is expected. If a team has few or no draws, the corresponding \(\sigma_i\) is weakly identified or tends toward very negative values, exactly as predicted by the theory of the model. The extreme negative values reported for teams such as Serbia, Iceland, Turkey, and Cyprus are consistent with this phenomenon and justify the use of penalized likelihood.

The official ranking is an administrative outcome determined by match points and tie-break rules. Our ranking is the ordering induced by the expected score in a virtual round robin against the whole field. These are conceptually distinct notions, and they are not supposed to coincide in every tournament.

This distinction is crucial for the comparison with Csató’s method. Csató’s least-squares framework also aims to reconstruct a more informative ranking from the incomplete comparison structure of a Swiss tournament, rather than merely accepting the official tie-break order. The difference is that his approach is based on a linear ranking procedure on the comparison graph, whereas ours starts from an explicit probabilistic model for win/draw/loss outcomes.

We now compare our ranking with the rankings reported by Csató. This comparison is especially meaningful because both approaches share a common ambition: to extract a more informative ordering from a Swiss-system event than the official lexicographic tie-breaks provide. However, they do so from two very different perspectives.

• Csató’s method is a graph-based least-squares ranking procedure.
• Our method is a maximum-likelihood statistical model with two latent player-specific dimensions.

Thus, even when the resulting rankings are close, their interpretation is not the same. Csató produces a robust ranking from incomplete pairwise comparisons; our model estimates latent strength and solidity, and derives the ranking as a consequence of the fitted probabilistic structure.

Case 2: 19th European Team Chess Championship 2013 Open section

chess-results.com/tnr114411.aspx

The overall agreement between methods, combined with systematic deviations from the official ranking, suggests that the issue lies in the tournament structure rather than in the choice of ranking method.

Case 3: IV Winter Chess Cosenza (Italy)

We finally consider a small Swiss-system tournament held in Italy (IV Winter Chess Cosenza), available at https://www.vegaresults.com/vega/7035.

Although this is not an elite event, it provides a particularly clear illustration of a structural issue that arises in many real-world tournaments.

The event consists of 48 players and only 5 rounds, resulting in an extremely sparse comparison structure. Each player faces only a very small subset of opponents, and the overall comparison graph is therefore highly incomplete.

In such a setting, the final standings are determined by aggregating results along a very short path of only five games. From a statistical perspective, this provides only a coarse and noisy estimate of the players’ relative strength.

A particularly instructive situation occurs in the final round: the two leading players meet with perfect scores. After player #2 loses this game, the official standings place player #6 in second position, despite the fact that player #2 had previously dominated most of the field.

This behaviour is not accidental. It reflects a fundamental feature of the Swiss system: the ranking is strongly influenced by the pairing path, which determines which comparisons are actually observed.

The table below reports the ranking induced by the Zermelo score:

The model confirms player #1 as the clear winner, consistent with the perfect score. However, it places player #2 in second position, ahead of player #3, and significantly above player #6, who was second in the official standings.

The key point is that player #2’s loss in the final round does not erase the information accumulated in the previous rounds. The probabilistic model integrates results across the entire comparison network and preserves this global information. By contrast, the official ranking effectively collapses all information into the final score and applies tie-breaks that depend on the realized pairing path.

This example highlights a general phenomenon:

• in short Swiss tournaments, the comparison graph is extremely sparse;
• the ranking becomes highly sensitive to individual results;
• the pairing process introduces structural biases in which comparisons are observed.

In particular, two players with similar scores may have faced opponents of very different strength, but this information is only partially captured by traditional tie-break systems.

Strategic implications of ranking criteria

An often overlooked consequence of ranking systems concerns the strategic behaviour they induce during the tournament.

In a Swiss system, the final standing is primarily determined by the total score, with tie-breaks applied only at the end. Since pairings depend on intermediate results, players are effectively navigating not only a competition, but also a dynamically evolving pairing structure.

This creates incentives for strategies that are not directly aligned with maximizing performance against the field.

Two typical phenomena illustrate this effect:

• Risk concentration: A player may choose to “play for everything” in a single critical game, knowing that a marginal score difference (e.g., half a point) can disproportionately affect the final standing.

• Strategic underperformance (so-called “Swiss gambit”): In some situations, it may be advantageous to concede or draw early games in order to avoid stronger opponents in subsequent rounds, while still aiming for a high final placement or prize.

These behaviours arise because the tournament outcome depends on a path-dependent sequence of pairings, rather than on a global evaluation of performance.

In contrast, model-based rankings—such as the Zermelo score or least-squares approaches—aggregate information across the entire comparison network.

In such frameworks:

• every game contributes to the estimation of player strength;
• early results are not “forgotten” or overridden by later outcomes;
• performance is evaluated globally, rather than through a sequence of local comparisons.

As a consequence, the incentives change fundamentally. There is no structural advantage in manipulating the pairing path, since all results contribute to a single global objective. Each game carries comparable informational weight, and improving performance at any stage of the tournament contributes positively to the final ranking.

This observation suggests that:

ranking systems based on a global estimation framework are more aligned with the sporting objective of the competition, as they reward consistent performance against the entire field rather than success along a particular pairing path.

In this sense, the choice of ranking method is not merely a technical detail, but has direct implications on the fairness and integrity of competitive behaviour.

Comparison with Bradley–Terry / Elo model

Consider the proposed bidimensional model with parameters \((r_i, \sigma_i)\). If the draw component is negligible, i.e. \[ e^{\sigma_i + \sigma_j} \ll e^{r_i}, e^{r_j}, \] then the model reduces to the classical Bradley–Terry formulation \[ P_{ij}^{W} \approx \frac{e^{r_i}}{e^{r_i} + e^{r_j}} = \frac{1}{1 + e^{-(r_i - r_j)}}, \]

\[ P_{ij}^{L} \approx \frac{e^{r_j}}{e^{r_i} + e^{r_j}} = \frac{1}{1 + e^{-(r_j - r_i)}}. \]

This is exactly the Bradley–Terry model. The Elo expected score is defined as \[ P_{ij}^{\text{Elo}} = \frac{1}{1 + 10^{-(R_i - R_j)/400}}. \]

Using the identity

\[ 10^x = e^{x \ln 10}, \] this can be rewritten as \[ P_{ij}^{\text{Elo}}=\frac{1}{1 + e^{-(R_i - R_j)\ln 10/400}}. \]

Comparing with the Bradley–Terry form \[ P_{ij} = \frac{1}{1 + e^{-(r_i - r_j)}}, \] we see that the two models coincide under the linear transformation \[ R_i = \beta \, r_i, \qquad \beta = \frac{400}{\ln 10}. \]

Hence: \[ P_{ij}^{\text{Elo}} = P_{ij}^{W} \] after rescaling. This shows that:

• when draws are negligible, the bidimensional model collapses to a one-dimensional logistic model;
• the classical Elo formulation is recovered exactly (up to scale);
• the proposed model is therefore a strict generalization of Bradley–Terry and Elo.

The agreement with Elo is not accidental. It reflects a structural property of the model: it naturally interpolates between a decisive regime (no draws) and a draw-dominated regime, without changing its functional form.

We illustrate the reduction to the Bradley–Terry / Elo model using players #1 and #2 from the Cosenza tournament. The estimated parameters are:

\[ r_1 = 22.9345,\quad \sigma_1 \approx 0, \]

\[ r_2 = 19.2220,\quad \sigma_2 = -0.1573. \]

Player #1 scored a perfect tournament (5/5), while player #2 scored 4/5, losing only in the final round against player #1.

Using the estimated parameters: \[ P_{12}^{W} \approx \frac{e^{r_1}}{e^{r_1} + e^{r_2}}=\frac{1}{1 + e^{-(r_1 - r_2)}} \approx 0.976. \]

The draw probability is:

\[ P_{12}^{D} =\frac{e^{\sigma_1+\sigma_2}}{Z_{12}} \approx 0, \]

Using the scaling

\[ R_i = \beta r_i, \quad \beta = \frac{400}{\ln 10} \approx 173.72, \]

we obtain:

\[ R_1 - R_2 \approx 645. \]

The Elo expected score is:

\[ P_{12}^{\text{Elo}}= \frac{1}{1 + 10^{-645/400}} \approx 0.976. \]

We now consider a single game from the final round of the Cosenza tournament, which ended in a draw.
We remove this game from the dataset and re-estimate the parameters \((r_i, \sigma_i)\). Using these updated parameters, we compute the probabilities of the three possible outcomes for the removed game. This demonstrates how the model assigns calibrated probabilities across all possible outcomes.

For the two players involved (#12 and #20), the estimated probabilities are:

\[ P_{20}^{W} \approx 0.84 \quad P^{D} \approx 0.16 \quad P^{D}_{12} \approx 0.001. \]

The model does not predict the exact outcome of the game, but assigns a non-negligible probability to a draw. This indicates that the observed result is consistent with the inferred structure, even when the game itself is not used in the estimation.

At the same time, the most likely outcome remains a win for the stronger player, reflecting the significant difference in strength. For comparison, we consider the one-dimensional Bradley–Terry component of the model, obtained by ignoring the draw term and using only the strength parameters \(u_i = e^{r_i}\). Under this reduction, the expected score of player #20 is approximately:

\[ E_{20}^{\text{Elo}} \approx 0.998, \]

which effectively rules out the possibility of a draw.

By contrast, the full bidimensional model assigns a substantial probability to a draw due to the interaction of the solidity parameters.

This example should not be interpreted as a statistically significant predictive test, since it involves a single observation in a small tournament. However, it provides a clear illustration of the qualitative difference between the models.

In particular:

• the proposed model can represent multiple outcome types (win, draw, loss);
• it assigns non-zero and interpretable probabilities to each outcome;
• it captures structural features (such as draw propensity) that are invisible in one-dimensional rating systems.

Thus, even in this simple setting, the example highlights how the model extends the descriptive capabilities of classical approaches.

Model Limitations and Perspectives

While the proposed bidimensional model provides a coherent and tractable framework for incorporating draws into paired-comparison models, several limitations should be acknowledged.

Identifiability and data sparsity: The model introduces two parameters per player, representing strength and draw propensity. In Swiss-system tournaments, where each participant plays only a limited number of games, the comparison graph is typically sparse. As a consequence, the second dimension (the solidity parameter) may be weakly identified for players with few or no draws. In particular, in the absence of draws, the model naturally collapses to the classical Zermelo formulation, while in tournaments with very high draw rates the strength component becomes less informative. This highlights that the effective dimensionality of the model is data-driven and may vary depending on the structure of the tournament. Despite this, empirical results suggest that, for typical tournament settings with a non-negligible proportion of decisive and drawn games, both dimensions are sufficiently informed by the data to yield stable and interpretable estimates.

Boundary behavior and regularization: The likelihood function may fail to attain a maximum in the interior of the parameter space, especially for players with extreme observed behavior (e.g., no draws). This issue is addressed through penalized likelihood, which ensures numerical stability and well-posed estimation. However, the choice of regularization parameters introduces an additional modeling component. While this approach is standard in statistical estimation and improves conditioning, a deeper analysis of its influence on parameter estimates and rankings remains an important topic for future work. In practice, we observe that moderate regularization leads to stable solutions without materially affecting the qualitative structure of the rankings.

Functional specification: The model assumes that draw probabilities arise from a multiplicative interaction between player-specific solidity parameters. This choice leads to a convenient and globally concave likelihood, but it is not the only possible specification. Alternative functional forms could potentially capture different aspects of draw behavior and may provide improved empirical performance in certain contexts. A systematic comparison between competing specifications, including model selection criteria and predictive accuracy, is left for future research.

Predictive validation: The empirical analysis presented in this paper focuses primarily on in-sample fit and structural interpretation. Although the model reproduces observed scores with high accuracy, further work is needed to assess its predictive performance on unseen data, for example through cross-validation or out-of-sample evaluation. Preliminary experiments suggest that the model behaves robustly across different tournaments, but a systematic validation study is required to quantify its generalization capabilities.

Ranking interpretation: The ranking induced by the model is based on expected scores in a virtual round robin. While this construction is natural from a probabilistic perspective, it represents one possible way of aggregating the bidimensional information into a total order. Alternative ranking criteria, possibly preserving more of the two-dimensional structure, could be explored. Nevertheless, the proposed ranking has the advantage of being directly derived from the probabilistic model and of providing a consistent global evaluation of performance.

These limitations do not undermine the internal consistency of the model, but rather delineate the scope within which its interpretation is most reliable. They also point to several directions for further development, including improved regularization strategies, alternative functional forms for draw modeling, and a more extensive empirical validation.

Overall, the empirical evidence presented in this paper indicates that the model captures meaningful structural features of tournament data and produces stable rankings across different settings. We view the present work as a first step toward a more comprehensive statistical treatment of chess tournaments with draws, and we expect that future research will refine and extend the framework proposed here.

Conclusion

We have introduced a probabilistic model for chess tournaments with draws based on a bidimensional latent representation of players. Each player is described by a pair (\(r_i,\sigma_i\)), capturing respectively competitive strength and draw propensity. Match outcomes arise naturally from the interaction of these two components, without the need for an exogenous draw parameter.

The model extends Zermelo’s formulation while preserving its structural simplicity and global concavity. From a statistical perspective, it provides a coherent likelihood-based framework with an explicit gradient and information matrix, enabling efficient estimation via Newton-type methods. The resulting parameters admit a clear interpretation and allow performance to be decomposed into dimensions that are conflated in traditional one-dimensional ratings.

A key outcome of the model is the Zermelo score, defined as the expected score in a virtual round robin against the entire field. This induces a global, model-consistent ranking that is independent of the particular pairing structure of the tournament.

The empirical analysis highlights a fundamental limitation of Swiss-system tournaments: rankings are derived from a sparse and path-dependent subset of all possible comparisons. As a result, final standings may depend strongly on the realized pairing sequence, while tie-break systems reweight existing information without introducing new statistical content.

In contrast, model-based approaches exploit the full comparison network and provide a global evaluation of performance. The consistency of the results with least-squares methods suggests that the observed discrepancies from official standings reflect structural properties of the tournament format rather than artifacts of a specific ranking technique.

More broadly, the results point to a natural separation of roles:

• the Swiss system as a pairing mechanism, designed to produce competitive rounds;
• statistical models as ranking tools, designed to infer player strength from incomplete comparisons.

This separation allows one to retain the practical advantages of the Swiss format while improving the statistical consistency and interpretability of tournament rankings.

References

• Zermelo, E. (1929).
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• Bradley, R. A., & Terry, M. E. (1952).
  Rank analysis of incomplete block designs: I. The method of paired comparisons.
  Biometrika, 39(3/4), 324–345.

• Davidson, R. R. (1970).
  On extending the Bradley–Terry model to accommodate ties in paired comparison experiments.
  Journal of the American Statistical Association, 65(329), 317–328.

• 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