Is there a formula to select tau if I have a specific uncertainty target in mind?

[joshua-revolutions]

Let's say I want players that have many games to have an uncertainty to be close to 2.5. Is there a formula I can use to know what value of tau I could use to achieve this result?

[vivekjoshy]

The way sigma is processed internally after a rating update is like so:

$$\sigma_{i} \leftarrow \sqrt{\sigma_{i}^2 + \tau^2}$$

Here is a good formula:

$$\tau = 2.5 \times \sqrt{|1 - 0.016 \times \sigma_{i}^2|}$$

In code:

import math

sigma = 0
tau = 2.5 * math.sqrt(abs(1 - 0.016 * sigma**2))
processed_sigma = math.sqrt(sigma**2 + tau**2)

At sigma of 0, it will always output 2.5 and at sigma of 25/2 it will output a value slightly higher than with the default tau of $\frac{25}{3}$.

Note: This has not been tested on simulations or real world data, so please let us know if it works. Thanks.