Measures of algorithmic bias can be roughly classified into four categories, distinguished by the conditional probabilistic dependencies to which they are sensitive. First, measures of "procedural bias" diagnose bias when the score returned by an algorithm is probabilistically dependent on a sensitive class variable (e.g. race or sex). Second, measures of "outcome bias" capture probabilistic dependence between class variables and the outcome for each subject (e.g. parole granted or loan denied). Third, measures of "behavior-relative error bias" capture probabilistic dependence between (...) class variables and the algorithmic score, conditional on target behaviors (e.g. recidivism or loan default). Fourth, measures of "score-relative error bias" capture probabilistic dependence between class variables and behavior, conditional on score. Several recent discussions have demonstrated a tradeoff between these different measures of algorithmic bias, and at least one recent paper has suggested conditions under which tradeoffs may be minimized. -/- In this paper we use the machinery of causal graphical models to show that, under standard assumptions, the underlying causal relations among variables forces some tradeoffs. We delineate a number of normative considerations that are encoded in different measures of bias, with reference to the philosophical literature on the wrongfulness of disparate treatment and disparate impact. While both kinds of error bias are nominally motivated by concern to avoid disparate impact, we argue that consideration of causal structures shows that these measures are better understood as complicated and unreliable measures of procedural biases (i.e. disparate treatment). Moreover, while procedural bias is indicative of disparate treatment, we show that the measure of procedural bias one ought to adopt is dependent on the account of the wrongfulness of disparate treatment one endorses. Finally, given that neither score-relative nor behavior-relative measures of error bias capture the relevant normative considerations, we suggest that error bias proper is best measured by score-based measures of accuracy, such as the Brier score. (shrink)

We map the recently proposed notions of algorithmic fairness to economic models of Equality of opportunity (EOP)---an extensively studied ideal of fairness in political philosophy. We formally show that through our conceptual mapping, many existing definition of algorithmic fairness, such as predictive value parity and equality of odds, can be interpreted as special cases of EOP. In this respect, our work serves as a unifying moral framework for understanding existing notions of algorithmic fairness. Most importantly, this framework allows us to (...) explicitly spell out the moral assumptions underlying each notion of fairness, and interpret recent fairness impossibility results in a new light. Last but not least and inspired by luck egalitarian models of EOP, we propose a new family of measures for algorithmic fairness. We illustrate our proposal empirically and show that employing a measure of algorithmic (un)fairness when its underlying moral assumptions are not satisfied, can have devastating consequences for the disadvantaged group's welfare. (shrink)