The introduction of randomization tests for the assessment of fairness is very useful, and the proposed method for encouraging fairness in an adversarial learning system is relevant and novel. The discussion phase showed that the paper would benefit in discussing this work in a broader context, beyond parity measures. First, to shortly describe the pros and cons of addressing fairness through these parity measures (also taking the caution words mentioned in the broader impact section). Second, this would allow to better contrast the randomization of the sensitive attribute that is carried out here with "interventions" on the sensitive features. In particular such as (i) a scheme where randomization would be simply used to mask the sensitive attribute, and (ii) a rudimentary assessment of counterfactual fairness that would be obtained by simply flipping the sensitive attribute. To quote a comment in the discussion: "It has been shown in [Counterfactual fairness, NeurIPS 2017], that intervening sensitive feature is not enough to mitigate unfairness, as there are other features that could be correlated with the sensitive feature (e.g. gender and marital status, race and zip code). So when intervening on the sensitive feature, we have to change also those non-sensitive features. More extensive review: Causal Reasoning for Algorithmic Fairness (in particular section 5.2.3 Equalised Odds and Calibration) by Loftus et al, 2018 https://arxiv.org/pdf/1805.05859.pdf. So in my opinion, there should be at least a discussion on this topic, and ideally a small evaluation on at least one of the datasets where causal graphs have been provided. The causal model for the two most frequently used datasets (Compas and Adult) have been derived: Razieh Nabi and Ilya Shpitser, Fair inference on outcomes, AAAI 2018, Figure 2." In addition, although the paper states early that the approach also applies when features contain (a proxy of) the sensitive attribute (X contains A), this point is not further developed in the paper; a didactic illustration could be useful. check for typos (Prop. 1, L. 121)