Fine-Grained Liquid Democracy for Cumulative Ballots

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, aiming at improving the quality of decision making, 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 voters' delegations. To this end we develop a theory based on 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. For example, we provide a parameterized algorithm whose running time depends on a distance from triviality of a given instance.