  Fairness has emerged as an important consideration in algorithmic
decision-making. Unfairness occurs when an agent with higher merit obtains a
worse outcome than an agent with lower merit. Our central point is that a
primary cause of unfairness is uncertainty. A principal or algorithm making
decisions never has access to the agents' true merit, and instead uses proxy
features that only imperfectly predict merit (e.g., GPA, star ratings,
recommendation letters). None of these ever fully capture an agent's merit; yet
existing approaches have mostly been defining fairness notions directly based
on observed features and outcomes.
  Our primary point is that it is more principled to acknowledge and model the
uncertainty explicitly. The role of observed features is to give rise to a
posterior distribution of the agents' merits. We use this viewpoint to define a
notion of approximate fairness in ranking. We call an algorithm $\phi$-fair
(for $\phi \in [0,1]$) if it has the following property for all agents $x$ and
all $k$: if agent $x$ is among the top $k$ agents with respect to merit with
probability at least $\rho$ (according to the posterior merit distribution),
then the algorithm places the agent among the top $k$ agents in its ranking
with probability at least $\phi \rho$.
  We show how to compute rankings that optimally trade off approximate fairness
against utility to the principal. In addition to the theoretical
characterization, we present an empirical analysis of the potential impact of
the approach in simulation studies. For real-world validation, we applied the
approach in the context of a paper recommendation system that we built and
fielded at the KDD 2020 conference.
