  Liquid democracy is a novel paradigm for collective decision-making that
gives agents the choice between casting a direct vote or delegating their vote
to another agent. We consider a generalization of the standard liquid democracy
setting by allowing agents to specify multiple potential delegates, together
with a preference ranking among them. This generalization increases the number
of possible delegation paths and enables higher participation rates because
fewer votes are lost due to delegation cycles or abstaining agents. In order to
implement this generalization of liquid democracy, we need to find a principled
way of choosing between multiple delegation paths. In this paper, we provide a
thorough axiomatic analysis of the space of delegation rules, i.e., functions
assigning a feasible delegation path to each delegating agent. In particular,
we prove axiomatic characterizations as well as an impossibility result for
delegation rules. We also analyze requirements on delegation rules that have
been suggested by practitioners, and introduce novel rules with attractive
properties. By performing an extensive experimental analysis on synthetic as
well as real-world data, we compare delegation rules with respect to several
quantitative criteria relating to the chosen paths and the resulting
distribution of voting power. Our experiments reveal that delegation rules can
be aligned on a spectrum reflecting an inherent trade-off between competing
objectives.

Computer Science and Game Theory