Impossibility results show that important fairness measures (independence, separation, sufficiency) cannot be satisfied at the same time under reasonable assumptions. This paper explores whether we can satisfy and/or improve these fairness measures simultaneously to a certain degree. We introduce information-theoretic formulations of the fairness measures and define degrees of fairness based on these formulations. The information-theoretic formulations suggest unexplored theoretical relations between the three fairness measures. In the experimental part, we use the information-theoretic expressions as regularizers to obtain fairness-regularized predictors for three standard datasets. Our experiments show that a) fairness regularization directly increases fairness measures, in line with existing work, and b) some fairness regularizations indirectly increase other fairness measures, as suggested by our theoretical findings. This establishes that it is possible to increase the degree to which some fairness measures are satisfied at the same time -- some fairness measures are gradually compatible.
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