Bayesian Tennis Ranking

A personal project to help my amateur tennis team field the lineup and win tournaments.

Let's consider a database of tennis matches as shown below:

	players A	players B	score	tournament
0	Donatello Luciani	Pellegrino Giovine	6-4 6-1	AICS 2023
1	Lorenzo Bellomo, Matteo Pardi	Marcello Cibin, Pierluigi Pacomio	6-1 6-1	AICS 2023
2	Pardi Marco	Lucio Conte	6-1 7-6	AICS 2023
3	Calogero Battelli	Uberto Oliboni	6-0 6-4	AICS 2023
4	Giuseppe Cavalcanti, Gioacchino Flaiano	Federico Gatto, Delfino Mozart	6-1 6-0	AICS 2023
...	...	...	...	...
379	Fulvio Zoppetti	Gianni Guarana	5-7 2-6	Mr. Dodo 22 - Fase Eliminatoria
380	Atenulf Solimena, Adriano Spinelli	Giacinto Orengo, Manuel Cannizzaro	5-7 3-6	Mr. Dodo 22 - Fase Eliminatoria
381	Gabriele Fantoni	Ennio Rizzoli	3-6 7-6 7-10	Mr. Dodo 22 - Fase Eliminatoria
382	Manuel Cannizzaro	Pasqual Dovara	6-3 7-6	Mr. Dodo 22 - Fase Eliminatoria
383	Donato Cattaneo, Sebastiano Alfieri	Ubaldo Ramazzotti, Gioffre Farina	6-3 6-3	Mr. Dodo 22 - Fase Eliminatoria

We aim to create a player ranking based on Bayesian statistics. The ranking will be structured as follows:

	player	ability
1	Giuseppe Cavalcanti	106.1
2	Cirillo Pisaroni	105.2
3	Gianluigi Caccianemico	104.9
4	Dionigi Vecellio	104.7
5	Carlo Peano	104.5
...	...	...
201	Sandro Abba	95.4
202	Marcello Cibin	95.2
203	Lazzaro Luna	94.8
204	Gionata Gulotta	94.4
205	Valerio Aporti	93.2

Traditional ranking methods are based on the 'you update when you play' rule:

• each time a player plays a match its ability changes
• the ability doesn't change if the player doesn't play (or decreases as player's inactivity increases)
• if a player wins its ability grows
• if a player lose its ability decreases

However, these methods have drawbacks that can be illustrated with examples:

1. There are five players: A, B, C, D, E. Let's say A defeats B, and then B defeats C, D and E. Since B won 3 matches and A just one, B will have a greater ability respect to A. But this is not what we would like to have, since A defeats B... A should be considered stronger than B.
2. Presently, A and B have the same ability, since they got similar results in their matches, played more or less with the same people. For a short period, A stops to play, while B continues. In this short period, B loses all matches he plays. This fact reveals that the ability of B was overrated: probably B had been lucky up to now in playing only against weak people. Thus, the ability of B decreases... but the ability of A remains the same. This is not what we would like to have, since we know that A and B should have similar abilities.

To address these issues, a new method based on Bayesian statistics is introduced. The method leverages a simple mathematical model for tennis matches.

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The Model

The model is straightforward. Each player $j = 1, \dots, M$ has an ability $a_j \in \mathbb{R}$. A tennis match between players A and B is represented by a sequence of independent points. Each point follows a Bernoulli distribution with a probability $\psi$, where $\psi$ is the probability of A winning the point. $\psi$ is determined by the players' abilities according to a logistic-like formula:

$$ \psi = \frac{1}{2} + \frac{1}{\pi} \arctan \left( \frac{a_A - a_B}{10} \right) \ . $$

In the case of a doubles match, the ability of a team will be defined as the mean of its players' abilities $a_A = (a'_A + a''_A)/2$. The formula's interpretation is aided by the following table:

$a_A - a_B$	$\psi$
$0$	$0.5$
$1$	$0.53$
$2$	$0.56$
$3$	$0.59$
$5$	$0.65$
$10$	$0.75$

More or less, we can say that $|a_A - a_B| \geq 3$ already determines the winner (if A wins a single point 60% of the time...). The model defines the likelihood of a match score $s$

$$ p(s|a_A, a_B) \ , $$

which can be derived analytically without particular difficulties (given the scoring system of the tournament we are interested to).

By treating the database as a collection of independent matches with scores $s_i$ where $i = 1, \dots, N$, the log-likelihood of the data is defined as:

$$ \log p(s_1, \dots, s_N | a_1, \dots, a_M) = \sum_{i=1}^{N} \log p (s_i | a_A^{(i)}, a_B^{(i)}) \ . $$

To estimate player abilities from the database, a Bayesian regularization term is introduced to the maximum likelihood estimator. A Gaussian distribution with mean 0 (to break the translational symmetry of the model during the optimization) and standard deviation $\sigma$ is proposed. Now: it's easy to show analytically that, under that gaussian distribution, $\langle |a_A - a_B| \rangle = 2 \sigma / \sqrt{\pi}$. Looking at the $a_A - a_B$ vs $\psi$ table above, a reasonable proposal for $\langle |a_A - a_B| \rangle$ might be 2, from which it follows $\sigma = \sqrt{\pi} \approx 1.77$. The loss function $\mathcal{L}$ to minimize becomes:

$$ \mathcal{L}(a_1, \dots, a_M) = -\log p(s_1, \dots, s_N | a_1, \dots, a_M) + \frac{1}{\pi} \sum_{j=1}^{M} {a_j}^2 \ . $$

To obtain the desired ranking, after the optimization the abilities are adjusted so that the median of the ranking is set to 100.

Temporal Weights

To account for the decreasing relevance of older matches, temporal weights $w_i$ are introduced into the log-likelihood:

$$ \log p(s_1, \dots, s_N | a_1, \dots, a_M) := \sum_{i=1}^{N} w_i \log p (s_i | a_A^{(i)}, a_B^{(i)}) \ . $$

A simple proposal for these weights is:

$$ w_i = 2^{- t_i / \tau} \ , $$

where $\tau$ represents the half-life of the exponential decay, and $t_i$ measures the time since the most recent match. A reasonable value for $\tau$ could be 10 months.

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Quick guide to the code

• Refer to main.ipynb for a quick tool usage.
• In the folder tennis_tools, README.md and quick_guide_notebook.ipynb introduce the main tool functionalities.
• In the folder tennis_tools/scoring_system, README.md guides custom tournament scoring system creation.

Contact Me

For any other additional information, you can email me at matteopardi2@gmail.com

Copyright

Copyright (C) 2023, Matteo Pardi

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <https://www.gnu.org/licenses/>.