In the field of algorithmic fairness, many fairness criteria have been proposed. Oftentimes, their proposal is only accompanied by a loose link to ideas from moral philosophy -- which makes it difficult to understand when the proposed criteria should be used to evaluate the fairness of a decision-making system. More recently, researchers have thus retroactively tried to tie existing fairness criteria to philosophical concepts. Group fairness criteria have typically been linked to egalitarianism, a theory of distributive justice. This makes it tempting to believe that fairness criteria mathematically represent ideals of distributive justice and this is indeed how they are typically portrayed. In this paper, we will discuss why the current approach of linking algorithmic fairness and distributive justice is too simplistic and, hence, insufficient. We argue that in the context of imperfect decision-making systems -- which is what we deal with in algorithmic fairness -- we should not only care about what the ideal distribution of benefits/harms among individuals would look like but also about how deviations from said ideal are distributed. Our claim is that algorithmic fairness is concerned with unfairness in these deviations. This requires us to rethink the way in which we, as algorithmic fairness researchers, view distributive justice and use fairness criteria.

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.