  Liquid democracy with ranked delegations is a novel voting scheme that unites
the practicability of representative democracy with the idealistic appeal of
direct democracy: Every voter decides between casting their vote on a question
at hand or delegating their voting weight to some other, trusted agent.
Delegations are transitive, and since voters may end up in a delegation cycle,
they are encouraged to indicate not only a single delegate, but a set of
potential delegates and a ranking among them. Based on the delegation
preferences of all voters, a delegation rule selects one representative per
voter. Previous work has revealed a trade-off between two properties of
delegation rules called anonymity and copy-robustness.
  To overcome this issue we study two fractional delegation rules: Mixed Borda
branching, which generalizes a rule satisfying copy-robustness, and the random
walk rule, which satisfies anonymity. Using the Markov chain tree theorem, we
show that the two rules are in fact equivalent, and simultaneously satisfy
generalized versions of the two properties. Combining the same theorem with
Fulkerson's algorithm, we develop a polynomial-time algorithm for computing the
outcome of the studied delegation rule. This algorithm is of independent
interest, having applications in semi-supervised learning and graph theory.
