A Story That Began Several Years Ago
Several years ago Roberto Ricca conducted an experimental study on tiebreak systems used in Swiss chess tournaments. At that time, most discussions among arbiters revolved around traditional criteria such as:
- Buchholz
- Sonneborn-Berger
- Direct Encounter
- and others
Ricca approached the problem differently. Instead of comparing tie-breaks only among themselves, he introduced the idea of global ranking criteria. The basic idea was simple:
evaluate tournament standings using models that consider the entire network of games.
Tie-break systems could then be judged by measuring how closely they reproduce the standings produced by these global models.
This was an important methodological step.
Global Ranking Models Considered
Among the global criteria examined in those experiments were two systems whose implementations had previously been discussed and coded:
- the Zermelo ranking model
- the Missing Points Score System (MPSS)
Both were implemented within the Vega program by the present author during earlier research on tournament ranking methods.
The Zermelo Model
The Zermelo model originates from the classical paper
Ernst Zermelo (1929) Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung.
In this model each player \( i \) is assigned a positive strength parameter \( u_i \). The probability that player \( i \) defeats player \( j \) is
$$ P_{ij}=\frac{u_i}{u_i+u_j}. $$
The parameters \(u_i\) are estimated by maximizing the likelihood of the observed tournament results. This produces a global estimate of player strengths consistent with all games of the tournament.
Missing Points Score System
The Missing Points Score System attempts to reconstruct the missing part of the result matrix of a Swiss tournament.
Since only a small fraction of all possible games is played, MPSS attempts to infer missing comparisons through transitivity. For instance
- A beats B
- B beats C
suggests that A should score against C.
Iterating this process approximates the results of a virtual round robin tournament. Although heuristic, this approach captures the idea that tournament results form a network of relations, not isolated matches.
An Interesting Empirical Observation
During those simulations Ricca observed that two of the ranking methods produced almost identical standings:
- Zermelo
- ABSELO
At the time this was presented as an empirical observation. However, the similarity was not surprising.
From the theory of rating systems it was already known that Elo-type models are closely related to Zermelo's probabilistic formulation.
Likelihood Formulation of the Zermelo Model
Suppose that between players \(i\) and \(j\):
- \(w_{ij}\) = number of wins of \(i\) over \(j\)
- \(w_{ji}\) = number of wins of \(j\) over \(i\)
and
$$ n_{ij}=w_{ij}+w_{ji} $$
is the total number of games between them (for the moment we ignore draws). The probability of observing exactly these results, up to a binomial coefficient, is
$$ L_{ij}= \left(\frac{u_i}{u_i+u_j}\right)^{w_{ij}} \left(\frac{u_j}{u_i+u_j}\right)^{w_{ji}}. $$
Multiplying over all pairs \(i<j\) gives the likelihood of the tournament:
$$ L(u_1,\dots,u_n)= \prod_{i<j} \left(\frac{u_i}{u_i+u_j}\right)^{w_{ij}} \left(\frac{u_j}{u_i+u_j}\right)^{w_{ji}}. $$
Taking logarithms:
$$ \ell(u)= \sum_{i<j} \left[ w_{ij}\log u_i + w_{ji}\log u_j - n_{ij}\log(u_i+u_j) \right]. $$
Maximum Likelihood Equations
Differentiating with respect to \(u_i\):
$$ \frac{\partial \ell}{\partial u_i} = \sum_{j\ne i} \left( \frac{w_{ij}}{u_i} - \frac{n_{ij}}{u_i+u_j} \right). $$
At the maximum likelihood point
$$ \frac{\partial \ell}{\partial u_i}=0. $$
Multiplying by \(u_i\):
$$ \sum_{j\ne i} w_{ij} = \sum_{j\ne i} n_{ij}\frac{u_i}{u_i+u_j}. $$
Define
$$ W_i=\sum_{j\ne i} w_{ij} $$
which represents the total number of wins of player \(i\). We obtain the equilibrium equations
$$ W_i= \sum_{j\ne i} n_{ij}\frac{u_i}{u_i+u_j}. $$
Thus for every player
observed wins = expected wins predicted by the model.
Iterative Form
The equilibrium equations can be written as
$$ W_i= u_i \sum_{j\ne i} \frac{n_{ij}}{u_i+u_j}. $$
Therefore
$$ u_i= \frac{W_i} {\displaystyle \sum_{j\ne i}\frac{n_{ij}}{u_i+u_j} }. $$
This equation leads directly to the classical iterative Zermelo algorithm.
Remark on Draws
The presentation above ignores draws and assumes only wins and losses.
In practice, chess games may end in draws. In Elo-type systems a draw is counted as half a point for each player.
This convention is equivalent to treating a draw as half a win and half a loss in the likelihood formulation, so the equilibrium equations remain unchanged.
Therefore the relationship between Zermelo and Elo derived above still holds when draws are present.
Logistic Form of the Model
Define ratings
$$ r_i=\alpha+\beta\log u_i. $$
Then
$$ u_i=e^{(r_i-\alpha)/\beta}. $$
Substituting into the probability formula:
$$ P_{ij} = \frac{u_i}{u_i+u_j} = \frac{1}{1+e^{-(r_i-r_j)/\beta}}. $$
Thus the Zermelo model is exactly a logistic expected score model written in different coordinates. The equilibrium equations can now be written in these new coordinates as
$$ W_i = \sum_{j\ne i} n_{ij} \frac{1}{1+e^{-(r_i-r_j)/\beta}}. $$
Relation with Elo
Consider the expected score of player \(i\)
$$ E_i(r)= \sum_{j\ne i} n_{ij} \frac{1}{1+e^{-(r_i-r_j)/\beta}}. $$
This is exactly the expected score formula used in Elo-type rating systems.
Suppose we perform a batch Elo update
$$ r_i^{(t+1)}= r_i^{(t)}+K\left(S_i-E_i(r^{(t)})\right), $$
where \(S_i\) is the observed score of player \(i\). In the no-draw case \(S_i=W_i\), so the equation reduces to the Zermelo equilibrium condition. If the iteration converges to a fixed point \(r^*\), then
$$ S_i=E_i(r^*). $$
But these are precisely the equilibrium equations obtained from the Zermelo maximum likelihood solution after the transformation
$$ r_i=\alpha+\beta\log u_i. $$
Thus the two procedures are closely related:
- Zermelo computes the maximum-likelihood solution directly.
- Elo performs an iterative correction of the ratings based on the difference between observed and expected scores.
If the process converges, both approaches lead to the same equilibrium ratings, up to an affine transformation of the rating scale.
Another derivation
The Zermelo model maximizes the log-likelihood
\[ \ell(u)= \sum_{i<j} \Big[ w_{ij}\log u_i + w_{ji}\log u_j - n_{ij}\log(u_i+u_j) \Big]. \]
For simplicity we use the logarithmic coordinate \(r_i=\log u_i\), which corresponds to fixing the scale parameter \(\beta=1\).
Then \(u_i=e^{r_i}\) and the log-likelihood becomes
\[ \ell(r)= \sum_{i<j} \Big[ w_{ij} r_i + w_{ji} r_j - n_{ij}\log\big(e^{r_i}+e^{r_j}\big) \Big]. \]
Now take the derivative with respect to \(r_i\). After a straightforward calculation we obtain
\[ \frac{\partial \ell}{\partial r_i} = S_i - E_i(r), \]
where \(S_i\) is the observed score of player \(i\) and
\[ E_i(r)= \sum_{j\ne i} n_{ij} \frac{1}{1+e^{-(r_i-r_j)}} \]
is the expected score predicted by the model.
Thus
gradient = observed score − expected score.
At the maximum-likelihood point we have
\[ \frac{\partial \ell}{\partial r_i}=0, \]
that is
\[ S_i = E_i(r). \]
These are exactly the equilibrium equations obtained earlier, written in logarithmic coordinates.
Now consider the Elo batch update
\[ r_i^{(t+1)} = r_i^{(t)} + K \big(S_i - E_i(r^{(t)})\big). \]
But we have just seen that
\[ S_i - E_i(r) = \frac{\partial \ell}{\partial r_i}. \]
Therefore
\[ r_i^{(t+1)} = r_i^{(t)} + K \frac{\partial \ell}{\partial r_i}. \]
In other words:
both methods are based on the same probabilistic model.
Zermelo computes the maximum-likelihood solution directly, while Elo performs a gradient ascent toward the same equilibrium.
Concluding Remarks
Ricca’s experimental work deserves recognition for an important conceptual insight.
By comparing traditional tie-break systems with global ranking models, he implicitly shifted the perspective from isolated criteria to the structure of the entire network of games played in a tournament.
From this viewpoint, tournament results form a statistical system in which each game contributes information about the relative strength of the players.
Global ranking models such as Zermelo or Elo exploit this structure by estimating player strengths using all the results simultaneously.
Traditional tie-break criteria, on the other hand, attempt to approximate the same information through simpler summary statistics based on partial aspects of the tournament (opponents’ scores, direct encounters, and similar quantities).
Seen in this light, many classical tie-break systems can be interpreted as coarse approximations of underlying global ranking principles.
This observation naturally raises a broader question.
If global models provide coherent strength estimates derived from the entire set of games, one may ask whether tournament standings should continue to rely on a collection of ad-hoc tie-break rules at all.
A natural alternative would be to derive the final ranking directly from a global statistical model of the tournament results.
This possibility will be explored in a subsequent article, where we will examine the historical tie-break criteria more closely and show how several of them can be understood as simplified projections of global ranking models.
Appendix
A. Logistic Scale
In the derivation above the expected score was written using the natural exponential
\[ P_{ij}=\frac{1}{1+e^{-(r_i-r_j)/\beta}}. \]
It is also common to write a similar formula using base 10:
\[ P_{ij}=\frac{1}{1+10^{-(R_i-R_j)/400}}. \]
Using the identity
\[ 10^x = e^{x\ln 10}, \]
the latter expression can be rewritten as
\[ P_{ij}=\frac{1}{1+e^{-(R_i-R_j)\ln 10/400}}. \]
Thus the two formulas are identical after the linear rescaling
\[ r_i=\frac{\ln 10}{400}R_i, \]
which corresponds to the logistic scale parameter
\[ \beta=\frac{400}{\ln 10}\approx 173.7. \]
B. Historical Note on the Elo Expected Score Formula
In Arpad Elo’s original work the expected score was not derived from a logistic model.
Instead, Elo assumed that the performance of a player in a single game follows a normal distribution centered at the player's rating.
Under this assumption the probability that player \(i\) scores against player \(j\) can be expressed through the cumulative normal distribution of the rating difference.
For practical reasons, however, the normal model is often approximated by the logistic function
\[ \frac{1}{1+10^{-x/400}}. \]
The logistic approximation is remarkably accurate: the maximum difference between the logistic curve with $\beta=\frac{400}{\ln 10}$ and the corresponding normal distribution model is only a few tenths of a percent in the relevant rating range.
From the perspective of comparison models, the logistic probability arises naturally from the maximum-likelihood formulation introduced by Zermelo (1929) and later formalized in the Bradley–Terry model.
Thus the practical Elo rating system is consistent with the probabilistic framework of maximum-likelihood ranking models.
versione 1 : 2026-03-12