A Appendix

A.1 On the ES fairness notion In this paper, we defined the ES fairness notion as follows, Pr {E Consider classifier R = r (X,A). A.4 Restating Theorem 5 for the statistical parity (SP) fairness notion Here we restate Theorem 5 for the statistical parity. The proof is similar to the proof of Theorem 5. Note that ( X,Y) and A are conditionally independent given A . Pr{r (X, 0) = ˆy |Y = 1,A = 0 } A.7 Numerical Experiment We compared EO and ES fairness notions in Table 2 after adding the following constraints to (13). Next, we prove the second part of the theorem.