A Note on the Significance Adjustment for FA*IR with Two Protected Groups

In this report we provide an improvement of the significance adjustment from the FA*IR algorithm of Zehlike et al., which did not work for very short rankings in combination with a low minimum proportion $p$ for the protected group. We show how the minimum number of protected candidates per ranking position can be calculated exactly and provide a mapping from the continuous space of significance levels ($\alpha$) to a discrete space of tables, which allows us to find $\alpha_c$ using a binary search heuristic.