  We consider binary group decision-making under a rich model of liquid
democracy recently proposed by Colley, Grandi, and Novaro (2022): agents submit
ranked delegation options, where each option may be a function of multiple
agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such
ballots are unravelled into a profile of direct votes by selecting one entry
from each ballot so as not to introduce cyclic dependencies. We study
delegation via monotonic Boolean functions, and two unravelling procedures:
MinSum, which minimises the sum of the ranks of the chosen entries, and its
egalitarian counterpart, MinMax. We provide complete computational dichotomies:
MinSum is hard to compute (and approximate) as soon as any non-trivial
functions are permitted, and polynomial otherwise; for MinMax the easiness
results extend to arbitrary-arity logical ORs and ANDs taken in isolation, but
not beyond. For the classic model of delegating to individual agents, we give
asymptotically near-tight algorithms for carrying out the two procedures and
efficient algorithms for finding optimal unravellings with the highest vote
count for a given alternative. These algorithms inspire novel tie-breaking
rules for the setup of voting to change a status quo. We then introduce a new
axiom, which can be viewed as a variant of the participation axiom, and use
algorithmic techniques developed earlier in the paper to show that it is
satisfied by MinSum and a lexicographic refinement of MinMax (but not MinMax
itself).
