  In this work, we consider the problem of intersectional group fairness in the
classification setting, where the objective is to learn discrimination-free
models in the presence of several intersecting sensitive groups. First, we
illustrate various shortcomings of existing fairness measures commonly used to
capture intersectional fairness. Then, we propose a new definition called the
$\alpha$-Intersectional Fairness, which combines the absolute and the relative
performance across sensitive groups and can be seen as a generalization of the
notion of differential fairness. We highlight several desirable properties of
the proposed definition and analyze its relation to other fairness measures.
Finally, we benchmark multiple popular in-processing fair machine learning
approaches using our new fairness definition and show that they do not achieve
any improvement over a simple baseline. Our results reveal that the increase in
fairness measured by previous definitions hides a "leveling down" effect, i.e.,
degrading the best performance over groups rather than improving the worst one.
