  We investigate efficient ways for the incorporation of liquid democracy into
election settings in which voters submit cumulative ballots, i.e., when each
voter is assigned a virtual coin that she can then distribute as she wishes
among the available election options. In particular, we are interested in
fine-grained liquid democracy, meaning that voters are able to designate a
partial coin to a set of election options and delegate the decision on how to
further split this partial coin among those election options to another voter
of her choice. The fact that we wish such delegations to be transitive --
combined with our aim at fully respecting such delegations -- means that
inconsistencies and cycles can occur, thus we set to find
computationally-efficient ways of resolving voter delegations. To this aim we
develop a theory based fixed-point theorems and mathematical programming
techniques and we show that for various variants of definitions regarding how
to resolve such transitive delegations, there is always a feasible resolution;
and we identify under which conditions such solutions are efficiently
computable.
