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\textit{Oh, \Section{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{}.  \Section{Irrational individuals?}  Theoretically, proxy voting has  an empty article!} interesting consequence: individual judgment sets (when including the proxy's judgment on each delegated issue) may turn out to be inconsistent.   As social choice theory (and judgment aggregation) standardly assume that individual preferences (and individual judgment sets), all the main results in those fields (i.e. Arrow's theorem) rely on this individual rationality assumption.   Allowing for inconsistent individual judgment sets hence opens a new level of generality in judgment aggregation.   Recall the doctrinal paradox, which famously shows that consistent individual judgment sets sometimes leads to an inconsistent collective outcome:  \begin{array}{lllll}  & p & q & p\land q \\  \hline  a & 1 & 0 & 0 \\  b & 0 & 1 & 0 \\  \hline  majority & 1 & 1 & 0 \\  \end{array}  p q p\land q  a 1 0 0  b 0 1 0  c 1 1 1  -----------------  G 1 1 0  In the more general setting of liquid democracy, inconsistent individual judgment sets may lead to consistent collective outcomes, for instance. This raises new research questions: what are the conditions for the collective outcome to be consistent? What are the criteria for rational delegation?  \section{Liquid Democracy as a limit case of DeGroot influence matrices}  First, look at liquid democracy as a kinda DeGroot style matrices for influence.   Look at simple non-probabilistic (0,1) case for an agenda with several issues.   The network of who delegates to whom is a forest (one tree for each issue).   Recall cases such as the following, with three agents, agenda {p,q, p \land q}, and the majority rule:  Despite the fact that all 3 individuals are consistent, such simple cases famously give rise to inconsistent collective choices.   The standard social choice perspective is to always take for granted individual rationality, and all typical results from social choice rely on this assumption.   However, consider the case of liquid democracy, where individuals may delegate differently for different issues (for instance, an agent may delegate on p but not on q). In that case, we may get an individual which is inconsistent. Hence, we might wanna drop the individual consistency assumption and get results for this more general case. (In other words, liquid democracy gives rationale for not imposing individual consistency).  This is an interesting case as it happens right now in real life (see examples in the above mentioned literature) !   ————————————————   II) Desirable properties for aggregation over networks?   What are the desirable properties (in a social choice like spirit) for aggregation over networks?   Look at the paper by Christian List’s on rules to generate profiles of opinions to profiles of opinions (instead of the typical judgment aggregation perspective: from profiles of opinions to unique opinion). This paper asks an interesting question and gives axioms for it, but the setting isn’t so nice, it merely renames social choice theoretic notions.   @Zoe: read last chapter of Davide’s primer (and article by List maybe)  ————————————————————   Additions 18 January:   So we can see delegating in the reversed perspective: if agents just copy the choices of their delegates. This can result in inconsistent individual choices. If we want to represent this kind of “copying” instead of “delegating”, then it would make more sense to   Can we use Degroot models to model liquid democracy?   Look for possibly existing generalization of the Degroot model.   @Zoe: look for the conditions for convergence for {0,1} case in the Jackson book.  ———————————————————  “bigger” questions for later:  under what conditions is the transformation determining an aggregation. Majority consistency is the most general property.   We take collective consistency to be a desirable property. So we want to check under what conditions liquid democracy would still preserve majority consistency, given the fact that we loose individual consistency.   Can we translate different kind of voting on networks into Degroot models?   For instance, choosing an option only if the unanimity of my neighbors have chosen it can be seen as delegating to all neighbors (and maybe one more step to treat this set of agents as one single delegate)  @Zoe: Check Kristof Apt’s work on coordination games on graphs  You can get started by \textbf{double clicking} this text block and begin editing. You can also click the \textbf{Text} button below to add new block elements. Or you can \textbf{drag and drop an image} right onto this text. Happy writing!