  Traditionally, recommender systems operate by returning a user a set of
items, ranked in order of estimated relevance to that user. In recent years,
methods relying on stochastic ordering have been developed to create "fairer"
rankings that reduce inequality in who or what is shown to users. Complete
randomization -- ordering candidate items randomly, independent of estimated
relevance -- is largely considered a baseline procedure that results in the
most equal distribution of exposure. In industry settings, recommender systems
often operate via a two-step process in which candidate items are first
produced using computationally inexpensive methods and then a full ranking
model is applied only to those candidates.
  In this paper, we consider the effects of inequality at the first step and
show that, paradoxically, complete randomization at the second step can result
in a higher degree of inequality relative to deterministic ordering of items by
estimated relevance scores. In light of this observation, we then propose a
simple post-processing algorithm in pursuit of reducing exposure inequality
that works both when candidate sets have a high level of imbalance and when
they do not. The efficacy of our method is illustrated on both simulated data
and a common benchmark data set used in studying fairness in recommender
systems.
