About 6 years ago, when we started to form the basic concepts of LiquidFeedback, we often heard people worrying about the problem of “circular delegations”: what happens if Alice delegates to Bob, Bob delegates to Chris, and Chris delegates to Alice? In our book, “The Principles of LiquidFeedback”, we explained that cyclic delegations are a nonexistent problem, because if all people in a cycle just delegate, then none of them will vote; while if one person casts a vote, the cycle will break automatically. [PLF, subsection 2.4.1 (“The myth of circular delegations”)]
The problem, however, is only nonexistent because if all people in a cycle delegate, none of the connected nodes in the delegation graph can unfold activities since any activity would break the cycle: as soon as somebody casts a vote, their (outgoing) delegation will be suspended (also compare Figures 2.6, 2.7, and 2.8 in [PLF]). We might assume that a revisitation of the previously dismissed issue of cyclic delegations is necessary if we extend the delegation model in such a way that (using a preference list) it is possible to delegate a decision to more than one person.
While there is probably an infinite number of possibilities to implement preferential delegation systems (and thus how to solve cyclic delegations), we were able to make a general statement about all preferential voting systems independently of the particular rules in effect to solve cyclic delegations. It could be shown that taking certain self-evident properties as given (Properties 2 through 5 in [PD]), any voting system will either treat directly and delegating voters unequally (violation of Property 6 in [PD]) or is prone to negative voting weight (violation of Property 7 in [PD]) if the voters' freely chosen delegates are respected (at least in trivial cases, see Property 1 in [PD]).
If we aim at treating all voters equally, we must ensure that direct and delegating voters are treated equally. [PLF, subsection 2.4.2] Following this political requirement, no matter which preferential delegation system we construct, it will be either susceptible to negative voting weight (violation of Property 7) or ignoring voters' preferences due to violation of Property 1 or disallowing voters to express their true preference if there is a cycle.
Circular delegations are still a nonexistent problem for the delegation model used by LiquidFeedback, but extending or replacing the model with preferential delegation will either lead to paradox situations in certain cases (negative voting weight, or disallowing voters to express their true preference) or break the important property of treating delegating and direct voters equally. It is important not to sacrifice the latter property, however, as it ensures equal opportunities for all participants. [PLF, subsection 2.4.2 (“Delegations and ‘one man – one vote’”)]