  Rankings have become the primary interface in two-sided online markets. Many
have noted that the rankings not only affect the satisfaction of the users
(e.g., customers, listeners, employers, travelers), but that the position in
the ranking allocates exposure -- and thus economic opportunity -- to the
ranked items (e.g., articles, products, songs, job seekers, restaurants,
hotels). This has raised questions of fairness to the items, and most existing
works have addressed fairness by explicitly linking item exposure to item
relevance. However, we argue that any particular choice of such a link function
may be difficult to defend, and we show that the resulting rankings can still
be unfair. To avoid these shortcomings, we develop a new axiomatic approach
that is rooted in principles of fair division. This not only avoids the need to
choose a link function, but also more meaningfully quantifies the impact on the
items beyond exposure. Our axioms of envy-freeness and dominance over uniform
ranking postulate that for a fair ranking policy every item should prefer their
own rank allocation over that of any other item, and that no item should be
actively disadvantaged by the rankings. To compute ranking policies that are
fair according to these axioms, we propose a new ranking objective related to
the Nash Social Welfare. We show that the solution has guarantees regarding its
envy-freeness, its dominance over uniform rankings for every item, and its
Pareto optimality. In contrast, we show that conventional exposure-based
fairness can produce large amounts of envy and have a highly disparate impact
on the items. Beyond these theoretical results, we illustrate empirically how
our framework controls the trade-off between impact-based individual item
fairness and user utility.
