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\section{An application to liquid democracy}  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'.\footnote{Under 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 This  is obviously a very too  constrained model, it does a model to  capture the very constrained form of realistic  opinion formation implicit in the so-called {\em liquid democracy} method for processes. We claim, however, that BDPs can serve as useful models to analyze aspects of  the opinion  aggregation of individual opinion. system called liquid democracy.  \subsection{Insights into liquid \subsection{Liquid  democracy} \subsection{Liquid In liquid  democracy (also known  as a BDP} ``proxy voting") for each issue submitted to vote, each agent can either cast his own vote, or he can delegate it to another agent (a ``proxy"), who he considers to be better equipped to make the best decision. As such, proxy voting stands in between direct democracy and standard representative democracy.\footnote{Liquid democracy is based on the software known as Liquid Feedback (\url{liquidfeedback.org}). Campaigns (e.g., Make Your Laws, \url{www.makeyourlaws.org}, US) and even parties with representatives that sat in national parliaments (e.g., Piratenpartei, Germany) are using and advocating the software.}  Liquid democracy is an aggregation procedure conceive to stand between direct and representative democracies.\footnote{Liquid democracy is supported by Despite  the software known as Liquid Feedback (\url{liquidfeedback.org}). Campaigns (e.g., Make Your Laws, \url{www.makeyourlaws.org}, US) and even parties with representatives that sat key ideas behind proxy voting have appeared  in national parliaments (e.g., Piratenpartei, Germany) are using and advocating the software.}   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 his own vote, or he can delegate it several contributions in political theory,  to another agent---a ``proxy")---and that agent can delegate date little systematic work exists on the system. Some analyses from a voting theory perspective have been put forth  in turn \cite{Green_Armytage_2014} but,  toyet another agent and so on. In  the end best of our knowledge, no model has been proposed so far.\footnote{The  only the agents that decided not to delegate their votes cast their ballots, for instance under majority rule. exception is possibly \cite{Boldi_2011}}  %Despite the key ideas behind proxy voting have appeared in several contributions in political theory, It has been argued that liquid democracy leads to collective outcomes which are closer  to %date little systematic work exists on the system. individual opinions than representative democracy \cite{}.   It has also been argued that liquid democracy would be too unstable to be implemented in practice as a replacement of representative democracy \cite{}.  Some analyses of liquid \subsection{Liquid  democracy from as  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 rather underinvestigated.  \subsection{Cycles}  Massive information loss (all votes in the cycles, and those upstream are lost) --> but not all cycles are necessarily disrupting the aggregation. BDP}