Bayesian Tennis Ranking

Motivation

Most traditional ranking methods work with a simple update rule:

• A player's ability changes only when they play.
• If they do not play, their ability either stays the same or decreases due to inactivity.
• When a player wins, their ability goes up, and when they lose, it goes down.

However, this approach can create some unexpected results. For example:

1. Imagine five players: A, B, C, D, and E. Player A beats B, but B then defeats C, D, and E. Since B has three wins while A has only one, traditional methods rank B higher than A. This seems unfair because A already proved they’re stronger by defeating B directly.
2. Now consider players A and B with similar abilities. A takes a break from playing, while B continues but loses every match. B’s rating drops as expected, but A’s stays the same. This suggests A is still strong, even though their ability should be seen as comparable to B’s, which has now been revealed as weaker.

To address these issues, we propose a Bayesian method that uses a simple mathematical model for tennis matches, offering a fairer way to calculate rankings.

Method

Each player $j=1,\dots ,M$ has an ability $a_j\in \mathbb{R}$.

When player A faces player B, the outcome of each point is modeled as a Bernoulli random variable with probability $\psi$, where:

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

For doubles, a team’s ability is the average of its two players, $a_A=(a_A^{\prime }+a_A^{″})/2$.

From the table below, we see how a difference in ability ($a_A-a_B$) affects the probability ($\psi$) of winning a point:

$a_A-a_B$	$\psi$
0	0.50
1	0.53
2	0.56
3	0.59
5	0.65
10	0.75

From the table, we can see:

• Differences close to zero indicate balanced players with roughly equal chances of winning.
• As the difference increases from one to three, the stronger player’s advantage grows steadily.
• Once the difference exceeds three, the outcome becomes predictable, as the stronger player is likely to win most points.

Likelihood and Regularization

If $s$ represents the score of a match between A and B, we can write its likelihood $p(s|a_A,a_B)$. For a database of $N$ matches with scores $s_i$, the log-likelihood is:

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

To regularize the solution, we add a prior distribution. Each ability $a_j$ is assumed to follow a Gaussian prior with mean zero (to break symmetry) and standard deviation $\sigma$. This prior ensures the optimization doesn’t produce extreme or unrealistic values. We choose $\sigma =\sqrt{\pi }$. This choice makes the expected difference between two players’ abilities about 2, which seems appropriate for a typical tournament based on the table above.

The loss function $\mathcal{L}$ to minimize is:

$$\mathcal{L}(a_1,\dots ,a_M)=-\log p(s_1,\dots ,s_N\mid a_1,\dots ,a_M)+\frac{1}{2\pi }\sum _{j=1}^M{a}_j^2.$$

After optimization, we shift and scale the final abilities so that the median rating is 100.

Temporal Weights

We can also reduce the impact of older matches by introducing weights $w_i$ into the log-likelihood:

$$\log p(s_1,\dots ,s_N\mid a_1,\dots ,a_M):=\sum _{i=1}^N{w}_i\log p(s_i\mid a_A^{(i)},a_B^{(i)}).$$

One way to define these weights is with an exponential decay:

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

where $t_i$ is the time since the most recent match, and $\tau$ is the half-life. A good default might be $\tau =8$ months.

Quick Guide

Please see A01. Tutorial for an example of how to use this package.

Contact Me

If you have any questions, feel free to email me at matteopardi2@gmail.com.

Copyright

Copyright (C) 2024, Matteo Pardi

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