In this section we give a glimpse of how BDPs interface with the collective decision-making system known as liquid democracy, and provide an interesting angle on one of its most discussed issues: delegation cycles.

The natural interpretation of BDPs is in terms of processes of opinion formation (cf. \cite{Grossi_2014}) where agents’ opinions are dictated by a personal ‘guru’.¹¹Under a convergence assumption, they could be thought of concrete instantiations of profile-transformation functions (from the set of all opinion profiles to the set of all opinion profiles) as studied in —\cite{List_2010}—. While this is obviously too constrained a model, it fits well with the sort of opinion formation process underpinning the aggregation method known as liquid democracy.

Liquid democracy²²Liquid democracy is based on the software known as Liquid Feedback (liquidfeedback.org). Campaigns (e.g., Make Your Laws, www.makeyourlaws.org, US) and even parties with representatives that sat in national parliaments (e.g., Piratenpartei, Germany) are using and advocating the software. is an aggregation method considered to stand between direct and representative democracy. At its heart is the so-called method of ‘proxy voting’ \cite{Miller_1969,Tullock_1992}. For each issue submitted to vote, each agent can either cast its own vote, or it can delegate its vote to another agent—a proxy—and that agent can delegate in turn to yet another agent and so on. Finally, the agents that decided not to delegate their votes cast their ballots (e.g., under majority rule). But their votes now carry as weight the number of all agents that entrusted them with their vote.

Analyses of liquid democracy from a social choice-theoretic perspective have been put forth in \cite{Alger_2006} and \cite{Green_Armytage_2014}, and from an algorithmic perspective in \cite{Boldi_2011}. However, the system remains fairly underinvestigated.

In proxy voting agents may decide to delegate their vote to exactly one other agent, or to exercise their voting right themselves. Delegations determine therefore a graph, and it should be clear that such graph has the same properties of the influence graphs studied in the earlier sections (they are serial and functional): every agent delegates to exactly one other agent (possibly itself) who becomes the agent’s trustee. So in proxy voting voters are effectively only those agents that entrusted themselves with the vote, and their vote carries as weight the cardinality of the set of all agents connected to them by a path in the delegation graph.

An equivalent narrative, often used by proponents of the liquid democracy system,³³ “While one way to describe delegations is the transfer of voting weight to another person, you can alternatively think of delegations as automated copying of the ballot of a trustee. While at assemblies with voting by a show of hands it is naturally possible to copy the vote of other people, in Liquid Democracy this becomes an intended principle” —\cite{liquid_feedback}—. is that each agent’s vote is actually a copy of the vote of its trustee. In this perspective, a proxy voting mechanism can be assimilated to a (converging) BDP. This perspective provides an interesting angle on the issue of delegation cycles.

Delegation cycles are seen by some as a key problem for proxy voting systems: if \(a\) delegates to \(b\), \(b\) to \(c\) and \(c\) to \(b\), what is the weight that a vote by any of the three agents will carry, and in general how will the opinions of these agents be counted? Proponents of liquid democracy tend to dismiss delegation cycles as a non-issue: since the three agents delegate their votes, none of them is casting a ballot and the cycles get resolved essentially by not counting the opinions of the agents involved in the cycle \cite{liquid_feedback}.⁴⁴Interestingly enough we could not find reference to the problem of delegation cycles in any of the recent work in social choice theory about proxy voting (—\cite{Alger_2006,Green_Armytage_2014}—). However, this seems problematic under the ‘vote copying’ reading of liquid democracy. In such a BDP perspective, this appears to be a drastic solution. The elimination of cycles not only hides to aggregation the opinions of the agents involved in cycles, but also the opinions of agents that may be linked to any of those agents by a delegation path. In other words information about entire connected components in the delegation graph may be lost.

Theorems \ref{theorem:opinion} and \ref{theorem:mu} offer an alternative solution by showing that not all cycles are necessarily bad news for convergence: cycles in which agents agree still support convergence of opinions, and therefore a feasible aggregation of opinions by proxy. This suggests that alternative proxy voting mechanisms could be designed to tolerate delegation cycles in which agents unanimously accept or reject a given issue, rather than simply discarding them, thereby obtaining an arguably better informed outcome of the aggregation.