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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}.} This While this  is obviouslytoo constrained  a model to very constrained model, it does  capture realistic the very constrained form of  opinion formation processes. We claim, however, that BDPs can serve as useful models to analyze aspects of implicit in  the opinion aggregation system called so-called {\em  liquid democracy. democracy} method for the aggregation of individual opinion.  \subsection{Liquid \subsection{Insights into liquid  democracy} In liquid \subsection{Liquid  democracy(also known  as ``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.} a BDP}  Despite the key ideas behind proxy voting have appeared in several contributions in political theory, Liquid democracy is an aggregation procedure conceive  to date little systematic work exists on stand between direct and representative democracies.\footnote{Liquid democracy is supported by  the system. Some analyses from a voting theory perspective have been put forth 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 \cite{Green_Armytage_2014} but, to national parliaments (e.g., Piratenpartei, Germany) are using and advocating the software.}   At its heart is  the best so-called method  of our knowledge, no model has been proposed ``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 to another agent---a ``proxy")---and that agent can delegate in turn to yet another agent and  so far.\footnote{The on. In the end  only exception is possibly \cite{Boldi_2011}} the agents that decided not to delegate their votes cast their ballots, for instance under majority rule.  It has been argued that liquid democracy leads to collective outcomes which are closer to individual opinions than representative democracy \cite{}.   It has also been argued that liquid democracy would be too unstable to be implemented %Despite the key ideas behind proxy voting have appeared  in practice as a replacement of representative democracy \cite{}. several contributions in political theory, to %date little systematic work exists on the system.  \subsection{Liquid Some analyses of liquid  democracy as from  a BDP} 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.