  Algorithmic Fairness is an established area of machine learning, willing to
reduce the influence of hidden bias in the data. Yet, despite its wide range of
applications, very few works consider the multi-class classification setting
from the fairness perspective. We focus on this question and extend the
definition of approximate fairness in the case of Demographic Parity to
multi-class classification. We specify the corresponding expressions of the
optimal fair classifiers. This suggests a plug-in data-driven procedure, for
which we establish theoretical guarantees. The enhanced estimator is proved to
mimic the behavior of the optimal rule both in terms of fairness and risk.
Notably, fairness guarantees are distribution-free. The approach is evaluated
on both synthetic and real datasets and reveals very effective in decision
making with a preset level of unfairness. In addition, our method is
competitive (if not better) with the state-of-the-art in binary and multi-class
tasks.
