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\section{Liquid Democracy} \section{Introduction}  %%%%%%%%%%%%%%%%%%%%%%%%%  \section{Preliminaries}  \subsection{Boolean DeGroot Processes}  In \cite{DeGroot}, DeGroot proposes a simple model of step by step opinion change under social influence. The model combines two types of matrices. Assuming a group of $n$ agents, a first $n\times n$ matrix represents the weighted influence network (who influences whom and how much), and a second $n \times m$ matrix represents the probability assigned by agents to $m$ different alternatives.   Both the agents' opinion and the influence weights are taken within $[0,1]$ and are (row) stochastic (each row sums up to $1$).  \begin{definition}  \ldots  \end{definition}  \subsection{Dynamics}  \subsection{Modal $\mu$-calculus}  %%%%%%%%%%%%%%%%%%%%  \section{Stabilization}  Limit cases of theorems by Jackson and the Toulousians.  %%%%%%%%%%%%%%%%%%%%  \section{An application to liquid democracy}  Liquid Democracy (or ``proxy voting") is a way of organizing voting. On 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.   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 in practice as a replacement of representative democracy \cite{}.  \begin{Conclusions}  %%%%%%%%%%%%%% NOTES %%%%%%%%%%%%%%%%%%%  \appendix  \section{Irrational individuals?}  Theoretically, proxy voting comes with an interesting feature: individual judgment sets (when including the proxy's judgment on each delegated issue) may turn out to be inconsistent.  
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\section{DeGroot influence}  In \cite{DeGroot}, DeGroot proposes a simple model of step by step opinion change under social influence. The model combines two types of matrices. Assuming a group of $n$ agents, a first $n\times n$ matrix represents the weighted influence network (who influences whom and how much), and a second $n \times m$ matrix represents the probability assigned by agents to $m$ different alternatives.   Both the agents' opinion and the influence weights are taken within $[0,1]$ and are (row) stochastic (each row sums up to $1$).   By multiplying the two matrices, a new opinion matrix is generated, representing the opinion state of agents after one influence step (for instance after they exchange information with each other). DeGroot, using the associativity of matrix multiplication, shows that whether the opinions of a group of agents converge in the limit depends on the underlying influence graph. If multiplying the influence matrix by itself converge in the limit, then any distribution of opinions of the agents will converge too.  %as Jackson puts it: no matter what beliefs p(0) the agents start with, they end up with limiting beliefs corresponding to the entries of $p(\infinite)=lim_t T_tp(0)$.   More precisely, convergence depends on (each connected and closed component) of the influence graph being aperiodic (that is, the greatest common divisor of all directed cycles lenghts is $1$.): ``convergence in overall society hold if an only if each closed and strongly connected set of nodes converges, which happens if and only if each such set is aperiodic." (jackson, p.233)