  Recent work highlights the role of causality in designing equitable
decision-making algorithms. It is not immediately clear, however, how existing
causal conceptions of fairness relate to one another, or what the consequences
are of using these definitions as design principles. Here, we first assemble
and categorize popular causal definitions of algorithmic fairness into two
broad families: (1) those that constrain the effects of decisions on
counterfactual disparities; and (2) those that constrain the effects of legally
protected characteristics, like race and gender, on decisions. We then show,
analytically and empirically, that both families of definitions \emph{almost
always} -- in a measure theoretic sense -- result in strongly Pareto dominated
decision policies, meaning there is an alternative, unconstrained policy
favored by every stakeholder with preferences drawn from a large, natural
class. For example, in the case of college admissions decisions, policies
constrained to satisfy causal fairness definitions would be disfavored by every
stakeholder with neutral or positive preferences for both academic preparedness
and diversity. Indeed, under a prominent definition of causal fairness, we
prove the resulting policies require admitting all students with the same
probability, regardless of academic qualifications or group membership. Our
results highlight formal limitations and potential adverse consequences of
common mathematical notions of causal fairness.
