  In this paper we extend the principle of proportional representation to
rankings. We consider the setting where alternatives need to be ranked based on
approval preferences. In this setting, proportional representation requires
that cohesive groups of voters are represented proportionally in each initial
segment of the ranking. Proportional rankings are desirable in situations where
initial segments of different lengths may be relevant, e.g., hiring decisions
(if it is unclear how many positions are to be filled), the presentation of
competing proposals on a liquid democracy platform (if it is unclear how many
proposals participants are taking into consideration), or recommender systems
(if a ranking has to accommodate different user types). We study the
proportional representation provided by several ranking methods and prove
theoretical guarantees. Furthermore, we experimentally evaluate these methods
and present preliminary evidence as to which methods are most suitable for
producing proportional rankings.
