1 Introduction
Liquid democracy [liquid_feedback] is a form of group decisionmaking considered to lie between direct and representative democracy. It has been used and popularized by campaigns for democratic reforms (e.g., Make Your Laws^{1}^{1}1www.makeyourlaws.org in the US) and parties (e.g., Demoex^{2}^{2}2demoex.se/en/ in Sweden, and Piratenpartei^{3}^{3}3www.piratenpartei.de in Germany), which used it to coordinate the behavior of party representatives in local as well as national assemblies. At its heart is voting via a delegable proxy, also called transferable or transitive proxy. 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. This differentiates liquid democracy from standard proxy voting [Miller_1969, Tullock_1992], where proxies cannot delegate their vote further. Finally, the agents that decided not to delegate their votes cast their ballots (e.g., under majority rule), but their votes now carry a weight consisting of the number of all agents that, directly or indirectly, entrusted them with their vote.
Context
Voting by delegable proxy was most probably first outlined in
[dodgson84principles]. Analyses of standard (nondelegable) proxy voting from a social choicetheoretic perspective—specifically through the theory of spatial voting—have been put forth in [Alger_2006] and [Green_Armytage_2014]. To date, little work has focused directly on liquid democracy: [kling15voting] provided an empirical study of voting behavior in liquid democracy based on election data from the Liquid Feedback^{4}^{4}4www.liquidfeedback.org platform of the German Piratenpartei; and [skowron16proportional] studied how, in the Liquid Feedback platform, issues to be submitted to vote are selected among usergenerated proposals via proportional rankings.^{5}^{5}5Another, somewhat tangential work is [Boldi_2011], which focused on algorithmic aspects of a variant of liquid democracy, called viscous democracy, with applications to recommender systems. However, to the best of our knowledge, no work has so far studied voting by delegable proxy as an aggregation rule in its own sake. We do this in the present paper, studying liquid democracy from the perspective of binary aggregation [Dokow_2010, grandi13lifting, Grossi_2014, endriss16judgment].Outline
The paper starts in Section 2 by introducing some preliminaries on the theory of binary aggregation. This preliminary section presents also novel results on binary aggregation with abstentions, which are needed for the analysis developed later in the paper. Section 3 introduces a simple model of liquid democracy based on binary aggregation. Section 4 establishes formal relations between the proposed model of liquid democracy and standard binary aggregation with abstentions. It studies the issue of circular delegations, and the issue of individual (ir)rationality when voting takes place on logically interdependent issues. The section finally moves from the analysis provided to outline two variants of delegable proxy, which: are more resilient against delegation cycles (Section 4.3); better preserve individual rationality when voting on logically interdependent issues (Section 4.4). Section 5 concludes.
2 Binary Aggregation
The formalism of choice for the analysis presented in this paper is binary aggregation with abstentions (see, for instance, [Dokow_2010]). This section is devoted to its introduction.
2.1 Opinions and Opinion Profiles
A binary aggregation structure (BA structure) is a tuple where:

is a nonempty finite set individuals ();

is a nonempty finite set of issues or propositions ();

is an (integrity) constraint, where is the propositional language constructed by closing under a functionally complete set of Boolean connectives (e.g., ).
A binary opinion is an assignment of acceptance/rejection values (or, truth values) to the set of issues . Allowing abstention amounts to considering incomplete opinions: an incomplete opinion is a partial function from to . We will study it as a function thereby explicitly denoting the undetermined value “” corresponding to abstention. Thus, (respectively, ) indicates that opinion rejects (respectively, accepts) the issue . Syntactically, the two opinions correspond to the truth of the literals or . For we write to denote one element from , and to denote , which we will refer to as the agenda of .
We say that the incomplete opinion of an agent is consistent if the set of formulas can be extended to a model of (in other words, if the set is satisfiable). Intuitively, the consistency of an incomplete opinion means that the integrity constraint is consistent with ’s opinion on the issues she does not abstain about. We also say that an incomplete opinion is closed whenever the following is the case: if the set of propositional formulas logically implies (respectively, ), then (respectively, ). That is, individual opinions are closed under logical consequence or, in other words, agents cannot abstain on issues whose acceptance or rejection is dictated by their expressed opinions on other issues. The set of incomplete opinions is denoted and the set of consistent and closed incomplete opinions . We will often refer to the latter simply as individual opinions, as they are the ones we focus on.
An opinion profile records the opinion on the elements of , of every individual in . Given a profile the projection is denoted (i.e., the opinion of agent in profile ). We also denote by the set of agents accepting issue in profile , by the set of agents rejecting in , and by the set of nonabstaining agents in . Sometimes we restrict the previous definitions to a coalition , so that (resp., ) denotes the set of agents in that accept (resp., reject) . Finally, we write to denote that the two profiles and are identical except, possibly, for the opinion of voter .
2.2 Aggregators
An aggregator is a function , from profiles of closed and consistent incomplete opinions to incomplete opinions. The issuebyissue strict majority rule () accepts an issue if and only if the majority of the nonabstaining voters accept that issue:
(1) 
We will refer to this rule simply as ‘majority’. Majority can be thought of as a quota rule. Quota rules in binary aggregation with abstentions are of the following form: accept when the proportion of nonabstaining individuals who accept is above the acceptancequota; reject when the proportion of nonabstaining individuals who reject is above the rejectionquota; and abstain otherwise:^{6}^{6}6There are several ways to think of quota rules with abstentions. Instead of a quota being a proportion of nonabstaining agents, one could for instance define rules with absolute quotas instead: accept when at least agents accept, independently of how many agents do not abstain. In practice, voting rules with abstention are often a combination of those two ideas: accept an issue if a big enough proportion of the population does not abstain, and if a big enough proportion of those accept it.
Definition 1 (Quota rules).
Let be a BA structure. A quota rule (for ) is defined as follows, for any issue , and any opinion profile :
(2) 
where is the cealing function. And, for , is a function assigning a positive rational number smaller or equal to to each issue, and such that, for each :
(3) 
A quota rule is called: uniform if, for all , ; it is called symmetric if, for all , .
Notice that the definition excludes trivial quota.^{7}^{7}7Those are quotas with value (always met) or (never met). Restricting to nontrivial quota is not essential but simplifies our exposition. It should also be clear that, by the constraint in (3), Definition 1 defines an aggregator of type as desired.^{8}^{8}8What needs to be avoided here is that both the acceptance and rejection quota are set so low as to make the rule output both the acceptance and the rejection of the issue. Notice finally that if the rule is symmetric, then (3) forces , for any given .
Example 1.
The majority rule (1) is a uniform and symmetric quota rule where and are set to meet the equation , for any issue and profile . This is achieved by setting the quota as , for each issue . More precisely one should therefore consider as a class of quota rules yielding the same collective opinions.
Example 2.
The uniform and symmetric unanimity rule is defined by setting . A uniform but asymmetric variant of unanimity can be obtained by setting and .
2.3 Properties of Agendas and Aggregators
Definition 2 (simple/evenly negatable agenda).
An agenda is said to be simple if there exists no set such that: , and is minimally inconsistent, that is:

is inconsistent with

For all , is consistent with (or, consistent).
An agenda is said to be evenly negatable if there exists a minimal inconsistent set such that for a set of even size, is consistent. It is said to be pathconnected if there exists such that where (conditional entailment) denotes that there exists , which is consistent with both and , and such that logically implies .
We refer the reader to [Grossi_2014, Ch. 2] for a detailed exposition of the above rather technical conditions. We provide just a simple illustrative example here.
Example 3.
Let and let . is not simple. The set is inconsistent with , but none of its subsets is. Let now and let . In this case, where issues are ordered by logical entailment, each minimally inconsistent set is of size , and the agenda is therefore simple. The trivial example of simple agenda is where , and the issues are therefore logically independent.
We proceed by recalling some wellknown properties of aggregators from the judgment and binary aggregation literatures, adapted to the setting of aggregation with abstention:^{9}^{9}9Such adaptation is, in many cases, nontrivial.
Definition 3.
Let be an aggregation structure. An aggregator is said to be:
 unanimous

iff for all , for all profiles and all : if for all , then . I.e., if everybody agrees on a value, that value is the collective value.
 anonymous

iff for any bijection , , where . I.e., permuting opinions among individuals does not affect the output of the aggregator.
 oligarchic

iff there exists (the oligarchs) s.t. and for any profile , and any value , iff for all . I.e., there exists a group of agents whose definite opinions always determine the group’s definite opinion on . If is oligarchic, with the same oligarchs on all issues , then it is called oligarchic.
 monotonic

iff, for all and all : for any profiles , if : (i) if and , then: if , then ; and (ii) if and , then: if , then . I.e., increasing support for a definite collective opinion does not change that collective opinion.
 independent

iff, for all , for any profiles : if for all , then . I.e., the collective opinion on each issue is determined only by the individual opinions on that issue.
 neutral

iff, for all , for any profile : if for all , , then . I.e., all issues are aggregated in the same manner.
 responsive

iff for all , there exist profiles such that and . I.e., the rule allows for an issue to be accepted for some profile, and rejected for some other.
 unbiased

iff for all , for any profiles : if for all , iff (we say that is the “reversed” profile of ), then iff . I.e., reversing all and only the individual opinions on (from acceptance to rejection and from rejection to acceptance) results in reversing the collective opinion on .
 rational

iff for any profile , is consistent and closed. I.e., the aggregator preserves the constraints on individual opinions.
Majority is unanimous, anonymous, monotonic, independent, neutral, responsive and unbiased, but it is not rational in general, as witnessed by wellknown judgment aggregation paradoxes (cf. [Grossi_2014]).
Finally, let us also define the following property. The undecisiveness of an aggregator on issue for a given aggregation structure is defined as the number of profiles which result in collective abstention on , that is:
(4) 
2.4 Some Results
Aggregation by majority is collectively rational under specific assumptions on the aggregation constraint:
Proposition 1.
Let be a BA structure with a simple agenda. Then is rational.
May’s theorem [May_1952] famously shows that for preference aggregation, the majority rule is in fact the only aggregator satisfying a specific set of desirable properties. A corresponding characterization of the majority rule is given in standard judgment aggregation (without abstentions): when the agenda is simple, the majority rule is the only aggregator which is rational, anonymous, monotonic and unbiased [Grossi_2014, Th. 3.2]. We give below a novel characterization theorem, which takes into account the possibility of abstentions both at the individual and at the collective level. To the best of our knowledge this is the first result of this kind in the literature on judgment and binary aggregation with abstention.
We first prove the following lemma:
Lemma 1.
Let be a uniform and symmetric quota rule for a given . The following holds: if and only if , for all .
That is, the quota rule(s) corresponding to the majority rule (Example 1) is precisely the rule that minimizes undecisiveness.
We can now state and prove the characterization result:
Theorem 1.
Let be an aggregator for a given . The following holds:

is a quota rule if and only if it is anonymous, independent, monotonic, and responsive;

is a uniform quota rule if and only if it is a neutral quota rule;

is a symmetric quota rule if and only if it is an unbiased quota rule;

is the majority rule if and only if it is a uniform symmetric quota rule which minimizes undecisiveness.
By the above theorem and Proposition 1, it follows that, on simple agendas, majority is the only rational aggregator which is also responsive, anonymous, systematic and monotonic.
We conclude by recollecting a wellknown impossibility result concerning binary aggregation with abstentions:
Theorem 2 ([Dokow_2010, Dietrich_2007]).
Let be a BA structure whose agenda is path connected and evenly negatable. Then if an aggregator is independent, unanimous and collectively rational, then it is oligarchic.
3 Binary Liquid Democracy
In binary aggregation with delegable proxy, agents either express an acceptance/rejection opinion or delegate the expression of such an opinion to another agent. The section models and studies this type of voting as a form of binary aggregation function.
3.1 Proxy Opinions, Profiles and Delegation Graphs
Let a BA structure be given and assume for now that , that is, all issues are logically independent. An opinion is an assignment of either a truth value or another agent to each issue in , such that (that is, selfdelegation is not an expressible opinion). We will later also require proxy opinions to be individually rational, in a precise sense (Section 4.1). For simplicity we are assuming that abstention is not a feasible opinion in proxy voting, but such assumption can be easily lifted in what follows.
We call functions of the above kind proxy opinions to distinguish them from standard (binary) opinions, and we denote by the set of all proxy opinions, the set of all individually rational proxy opinions (as defined later in Section 4.1). Finally, denotes the set of all profiles of proxy opinions, which we call, proxy profiles.
Each proxy profile induces a delegation graph where for :
(5) 
The expression stands for “ delegates her vote to on issue ”. Each is a socalled functional relation. It corresponds to the graph of an endomap on . So we will sometimes refer to the endomap of which is the graph. Relations have a very specific structure and can be thought of as a set of (converging) trees whose roots all belong to cycles (possibly loops).
The weight of an agent w.r.t. in a delegation graph is given by its indegree with respect to (i.e., the reflexive and transitive closure of ):^{10}^{10}10 We recall that the reflexive transitive closure of a binary relation is the smallest reflexive and transitive relation that contains . . The weight of a coalition is defined naturally as . This definition of weight makes sure that each individual carries the same weight, independently of the structure of the delegation graph. Alternative definitions of weight are of course possible.
For all , we consider the function defined as The function associates to each agent (for a given issue ), the (singleton consisting of the) last agent reachable from via a path of delegation on issue , when it exists (and otherwise). Slightly abusing notation we will use to denote an agent, that is, the guru of over when . If we call a guru for . Notice that iff , i.e., is a guru of iff it is a fixpoint of the endomap .
If the delegation graph of a proxy profile is such that, for some , there exists no such that is a guru of , we say that graph (and profile ) is void on . Intuitively, a void profile is a profile where no voter expresses an opinion, because every voter delegates her vote to somebody else.
Given a BA structure , a proxy aggregation rule (or proxy aggregator) for is a function that maps every proxy profile to one collective incomplete opinion. As above, denotes the outcome of the aggregation on issue .
3.2 Proxy Aggregators
The most natural form of voting via delegable proxy is a proxy version of the majority rule we discussed in Section 2:^{11}^{11}11On the importance of majority decisions in the current implementation of liquid democracy by Liquid Feedback cf. [liquid_feedback, p.106].
(6) 
Again, the notation (resp., ) denotes the set of voters accepting (resp., rejecting) in proxy profile . Intuitively, an issue is accepted by proxy majority in profile if the sum of the weights of the agents who accept in exceeds the majority quota, it is rejected if the sum of the weights of the agents who reject in exceeds the majority quota, and it is undecided otherwise. Note that (and similarly for ), that is, the sum of the weights of the gurus accepting (rejecting) is precisely the cardinality of the set of agents whose gurus accept (reject) .
It should be clear that for any quota rule a proxy variant of can be defined via an obvious adaptation of (6).
4 Analysis and Extensions
In this section we provide an analysis of liquid democracy by highlighting two issues—the failure of rationality in ballots under delegable proxy voting, and the occurrence of delegation cycles—and by embedding it in the theory of binary aggregation with abstentions presented in Section 2. We also advance proposals for simple modifications of the delegable proxy voting method in order to address the issues we identify.
4.1 Individual and Collective Rationality
In our discussion so far we have glossed over the issue of logically interdependent issues and collective rationality. The reason is that under the delegative interpretation of liquid democracy developed in the previous sections individual rationality itself appears to be a more debatable requirement than it normally is in classical aggregation.
A proxy opinion is individually rational if the set of formulas
(7) 
is satisfiable (consistency), and if whenever (7) entails , then belongs to it (closure). That is, the integrity constraint is consistent with ’s opinion on the issues she does not delegate on, and the opinions of her gurus (if they exist), and those opinions, taken together, are closed under logical consequence.
The consistency and closure of (7) capture a highly idealized way of how delegation works: voters are assumed to be able to check or monitor how their gurus are going to vote, and always modify their delegations if an inconsistency arises. So the constraint appears highly unrealistic under a delegative interpretation of liquid democracy. Aggregation via delegable proxy has at least the potential to represent individual opinions as irrational (inconsistent and/or not logically closed).
The assumption of individual rationality for proxy opinions, however, is needed in order to establish variants of known binary aggregation results for the case of liquid democracy, to which we turn now.
4.2 Embedding
Having defined individual rationality in the previous section, it is possible now to study embeddings from proxy voting to standard aggregation, and vice versa.
Aggregation in liquid democracy—as conceived in [liquid_feedback]—should satisfy the principle that the opinion of every voter, whether expressed directly or through proxy, should be given the same weight.^{12}^{12}12 “[…] in fact every eligible voter has still exactly one vote […] unrestricted transitive delegations are an integral part of Liquid Democracy. […] Unrestricted transitive delegations are treating delegating voters and direct voters equally, which is most democratic and empowers those who could not organize themselves otherwise” [liquid_feedback, p.3436] In other words, this principle suggests that aggregation via delegable proxy should actually be ‘blind’ for the specific type of delegation graph arising. Making this more formal, we can think of the above principle as suggesting that the only relevant content of a proxy profile is its translation into a standard opinion profile (with abstentions) via a function defined as follows: for any and , if (i.e., if has a guru for ), and otherwise. Clearly, if we assume proxy profiles to be individually rational, the translation will map proxy opinions into individually rational (consistent and closed) incomplete opinions. By extension, we will denote by the incomplete opinion profile resulting from translating the individual opinions of a proxy profile .
The above discussion suggests the definition of the following property of proxy aggregators: a proxy aggregator has the one man–one vote property (or is a one man–one vote aggregator) if and only if for some aggregator (assuming the individual rationality of proxy profiles).^{13}^{13}13Not every proxy aggregator satisfies the one man–one vote property. By means of example, consider an aggregator that uses the following notion of weight accrued by gurus in a delegation graph. The weight of is where denotes the length of the delegation path linking to . This definition of weight is such that the contribution of voters decreases as their distance from the guru increases. Aggregators of this type are studied in [Boldi_2011]. The class of one man–one vote aggregators can therefore be studied simply as the concatenation where is an aggregator for binary voting with abstentions, as depicted in Figure 1 (left).
Example 4.
Proxy majority (6) is a one man–one vote rule aggregator. It is easy to check that, for any proxy profile : .
It follows that for every proxy aggregator the axiomatic machinery developed for standard aggregators can be directly tapped into. Characterization results then extend effortlessly. In particular, Theorem 1 implies the following:
Fact 1 (Characterization of proxy majority).
A one man–one vote proxy aggregator for a given is proxy majority iff is anonymous, independent, monotonic, responsive, neutral and minimizes undecisiveness.
The fact may well be considered as a theoretical argument in favor of the use of proxy majority in aggregation with delegable proxy as currently done, for instance, in the Liquid Feedback platform.
Similarly, we can study an embedding of standard aggregation into voting with delegable proxy. For example, we can define a function from opinion profiles to individually rational proxy profiles as follows. For a given opinion profile , and issue consider the set of individuals that abstain in and take an enumeration of its elements, with . The function is defined as follows: for any and , if , , otherwise.^{14}^{14}14Notice that since selfdelegation (that is, ) is not feasible in proxy opinions, this definition of works for profiles where, on each issue, either nobody abstains or at least two individuals abstain. Clearly, a dummy abstaining voter can then be added in profiles where only one individual abstains. A translation of this type allows to think of standard aggregators as the concatenation , for some proxy aggregator , as in Figure 1 (right). The following impossibility result for aggregation with delegable proxy voting can then be obtained as a direct consequence of Theorem 2:
Fact 2.
Let be such that its agenda is path connected and evenly negatable. For any proxy aggregator , if is independent, unanimous and collectively rational, then it is oligarchic.
4.2.1 Cycles and Abstentions
Proxy aggregators rely on the existence of gurus in the underlying delegation graphs. If the delegation graph on issue contains no guru, then the aggregator has access to no information in terms of who accepts and who rejects issue . To avoid bias in favor of acceptance or rejection, such situations should therefore result in an undecided collective opinion. That is for instance the case of . However, such situations may well be considered problematic, and the natural question arises therefore of how likely they are, at least in principle.
Proposition 2.
Let be a BA structure where (i.e., issues are independent) and fix an issue . If each proxy profile is equally probable (impartial culture assumption), then the probability that a given proxy profile is such that is a profile in which every voter abstains tends to as tends to infinity.
It follows that for unanimous and one man–one vote proxy aggregators, asymptotically, there is a considerable chance that a profile results in collective abstention. Now contrast this with the probability that all agents abstain on an issue when each voter either expresses a or opinion or abstains (that is, the binary aggregation with abstentions setting studied earlier). In that case the probability that everybody abstains, and therefore the profile is void, clearly tends to as tends to infinity.
Proposition 2
should obviously not be taken as a realistic estimate of the effect of cycles on collective abstention, moreover concrete implementations of delegable proxy voting may be designed to detect and resolve cycles (cf.
[yamakawa07toward, kling15voting]). Ultimately, theoretical (e.g., game theoretic) models of delegation behavior in voters or, ideally, election data should be used to assess whether delegation cycles ever lead large parts of the electorate to effectively lose representation in the aggregation mechanism. Still, the link we highlight between delegable proxy and collective abstention is, to the best of our knowledge, novel and has escaped so far recognition within the liquid democracy literature.^{15}^{15}15Delegation cycles are normally criticized for the wrong reason, that is, the fact that hey may be interpreted as to lead to an infinite accrual of voting power: “The by far most discussed issue is the socalled circular delegation problem. What happens if the transitive delegations lead to a cycle, e.g. Alice delegates to Bob, Bob delegates to Chris, and Chris delegates to Alice? Would this lead to an infinite voting weight? Do we need to take special measures to prohibit such a situation? In fact, this is a nonexistent problem: A cycle only exists as long as there is no activity in the cycle in which case the cycle has no effect. As already explained […], as soon as somebody casts a vote, their (outgoing) delegation will be suspended. Therefore, the cycle naturally disappears before it is used. In our example: If Alice and Chris decide to vote, then Alice will no longer delegate to Bob, and Chris will no longer delegate to Alice […]. If only Alice decides to vote, then only Alice’s delegation to Bob is suspended and Alice would use a voting weight of 3. In either case the cycle is automatically resolved and the total voting weight used is 3.” [liquid_feedback, Section 2.4.1] Cf. [Behrens15]. We agree that the alleged accrual of infinite voting power is immaterial. However, the fact that the occurrence of a cycle leads to the loss of representation of the voters in the cycle—and of those delegating to them—does not seem to have yet been acknowledged.4.3 Delegable Proxy with Default Values
Motivated by the above analysis, we outline a simple modification of voting via delegable proxy, which requires agents to always submit a substantive opinion on the issues, and at the same time indicate a trustee. In this view, an opinion (called proxy opinion with default) is therefore a function assigning to every issue an acceptance or rejection value and, at the same time, an individual, which is to be considered the individual the vote is delegated to. Intuitively, each voter expresses an opinion but accepts that opinion to be overruled by the opinion of the individual she entrusts. Note that such individual may well be the voter herself (e.g., ). We refer to profiles of such opinions as proxy profiles with default.
Let denote the set of cycles of the delegation graph such that among the agents in the cycle there exists a majority accepting . The set is defined in the symmetric way. Now define proxy majority as an aggregator for profiles of proxy opinions with default values:
(8) 
where, recall, is the cumulative weight (w.r.t. ) of the agents in . The intuition behind (8) is to use each cycle, and not only loops (i.e., gurus), as sources of information for the proxy aggregator, by attributing to the individuals in a cycle the majority default opinion present in that cycle.
As one might intuitively expect, this is enough to break the link between delegation cycles and group abstention we identified with Proposition 2. To state the following result we need to adapt the translation function for proxy profiles, to a translation function translating proxy profiles with default to opinion profiles with abstentions: for any and , where is the cycle reachable from via .
Proposition 3.
Let be a BA structure where (i.e., issues are independent) and fix an issue . If each proxy profile with default is equally probable (impartial culture assumption), then the probability that a given proxy profile with default is such that is a profile in which every voter abstains tends to as tends to infinity.
4.4 Individually Rational Delegable Proxy
Delegable proxy voting can also be studied from a different perspective. Imagine a group where, for each issue , each agent copies the binary— or —opinion of a unique trustee.^{16}^{16}16For simplicity, in this section we assume agents are therefore not allowed to abstain, although this is not a crucial assumption for the development of our analysis. Imagine that this group does so repeatedly until all agents (possibly) reach a stable opinion. These new stable opinions can then be aggregated as the ‘true’ opinions of the individuals in the group, for instance, via majority. The collective opinion of a group of agents, who either express a binary opinion or delegate it to another agent, is (for one man–one vote proxy aggregators) the same as the output obtained from a vote where each individual has to express a binary opinion but gets there by copying the opinion of her trustee (possibly the agent itself). In this perspective, aggregation via delegable proxy can be assimilated to a (stabilizing) process of opinion formation on delegation graphs.
The above interpretation of liquid democracy is explicitly put forth in [liquid_feedback].^{17}^{17}17 “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” [liquid_feedback, p. 22]. Under this ‘votecopying’ interpretation, the constraint on individual rationality—consistency and closure of (7)—is, arguably, more easily defendable: each agent will copy opinions coming from her trustees only if consistency and closure are preserved.
4.4.1 Boolean DeGroot Processes
We briefly develop the above intuition, outlining an opinion diffusion model of delegable proxy which preserves individual rationality in a natural way.^{18}^{18}18As we will consider just binary opinions (without abstentions), the concept of individual rationality can be slightly simplified: requiring an opinion to be consistent suffices as in the case of binary opinions without abstentions, consistency implies closedness.
Definition 4.
Fix a BA structure , a profile of consistent binary opinions, and a delegation graph . Consider the stream of opinion profiles recursively defined as follows:

Base:

Step: for all , ,
where .
When is set to , the above defines independent binary processes, one for each issue . Each of such processes is a Boolean extremal case of a DeGroot stochastic process [Degroot_1974] where opinions are binary, and each agent can trust one and at most one other agent. When the constraint is not a tautology, the definition guarantees that at each step individual opinions remain consistent with . We call processes defined by the above dynamics individually rational Boolean DeGroot processes (in short, BDPs).^{19}^{19}19Other types of dynamics are of course possible. A recent systematic investigation of opinion diffusion on logically interdependent issues is [Botan16]. For a broader study of Boolean DeGroot processes in the context of models of binary opinion diffusion on networks we refer the reader to [christoff17stability].
4.4.2 Stabilization
We say that the stream of opinion profiles stabilizes if there exists such that for all , if , then . We call such profile the limit profile. A BDP that stabilizes can be thought of as an opinion transformation function [List_2010] turning an initial binary profile into a new binary profile equal to the limit profile. In this view, individually rational proxy aggregation consists first in an opinion transformation, implemented through a BDP, and then the application of an aggregator (e.g., ) on the profile of transformed opinions . A BDP that does not converge, can similarly be thought of as mapping the initial profile to a profile involving some level of abstention, where agents connected to some delegation cycle may not end up stabilizing and are therefore considered to abstain. We conclude by establishing conditions for individually rational Boolean DeGroot processes to stabilize.
Theorem 3.
Fix a BA structure , a profile of consistent (w.r.t. ) binary opinions, and a delegation graph . Then the following holds: if for all , for all such that is a cycle in , and all : , then the individually rational BDP (for , and ) stabilizes in at most steps, where .
When , the opposite direction also holds, and one can obtain a characterization of the notion of stabilization for BDPs based on properties of the initial opinion profile and of the delegation graph.
Theorem 4.
Fix a BA structure , a profile of consistent (w.r.t. ) binary opinions, and a delegation graph , and let . Then the following statements are equivalent:

The BDP (for and G) stabilizes.

For all , there is no set of agents such that: is a cycle in and there are two agents such that .
A special case of Theorem 4 is the case in which contains no cycle of length . In such case, a direct consequence of the theorem is that the process stabilizes from any profile. This is also a corollary of a known stabilization result for DeGroot processes (cf. [jackson08social, p.233]).
5 Conclusions
The paper has shown how delegable proxy voting (liquid democracy) can be understood as an aggregator within the theory of binary aggregation with abstentions, for which we provided a novel characterization theorem of issuewise majority (Theorem 1). This has allowed us to clarify the impact of cyclical delegations on individual and collective abstentions (Proposition 2) and to suggest alternative aggregators requiring individuals to reveal a default opinion, which can be shown to better behave in the presence of delegation cycles (Proposition 3). Finally we showed how delegable proxy interferes with individual rationality, a standard tenet of social choice theory. Also in this case we showed how liquid democracy could be adjusted—in the form of a stabilizing diffusion process—in order to preserve individual rationality (Theorem 3).
Proofs
Proof of Proposition 1.
If the agenda is simple, then all minimally inconsistent sets have cardinality , that is, are of the form such that for . W.l.o.g. assume and . Suppose towards a contradiction that there exists a profile such that is inconsistent, that is, , and . By the definition of (1) it follows that and . Since by assumption, and since individual opinions are consistent and closed, and . From the fact that we can thus conclude that . Contradiction. ∎
Proof of Lemma 1.
We establish the claim through a series of equivalences. Observe first of all that a uniform and symmetric quota rule is such that (a) , for all if and only if, (b) for any and , if and only if , that is, an even number of voters vote and the group is split in half. Now, (b) is the case if and only if, (c) the quota of are set in such a way that for any profile and issue . In turn (c) is the case if and only if, (d) the quota of are set as , which are the quota defining (Example 1). ∎
Proof of Theorem 1.
Claim 1 Lefttoright: Easily checked. Righttoleft: Let be an anonymous, independent, monotonic, and responsive aggregator. By anonymity and independence, for any , and any , the only information determining the value of are the integers and . By responsiveness, there exists a nonempty set of profiles . Pick to be any profile in with a minimal value of and call this value . Now let be any profile such that and . This implies that and . By monotonicity, it follows that . By iterating this argument a finite number of times we conclude that whenever , we have that . Given that was defined as a minimal value, we conclude also that if , then . The argument for is identical.
Proof of Proposition 2.
The claim amounts to computing the probability that a random proxy profile induces a delegation graph that does not contain gurus (or equivalently, whose endomap has no fixpoints) as tends to infinity. Now, for each agent , the number of possible opinions on a given issue (that is, functions ) is (recall cannot express “” as an opinion). The number of opinions in which is delegating her vote is . So, the probability that a random opinion of about is an opinion delegating ’s vote is . Hence the probability that a random profile consists only of delegated votes (no gurus), for a fixed issue, is . The claimed value is then established through this series of equations:
This completes the proof. ∎
Proof of Proposition 3.
The claim amounts to computing the probability that a random proxy profile with default opinions induces a delegation graph (equivalently, an endomap ) whose cycles are all hung majorities, that is, whose cycles are all even and exactly half the agents in each cycle accept . As opinion with defaults consist of both a value and a trustee we can treat the probability of each component as independent: the number of all possible proxy profiles with default opinions is, therefore, . First of all, recall that a delegation graph can be represented as a set of trees whose roots are nodes in a cycle, that is, as trees whose roots are elements of a permutation of a subset of . The number of ways of arranging elements in trees rooted on elements (with ) is given by the following recursive function (cf. [purdom68cycle]):
(9) 
with and for any . So the number of all possible delegation graphs equals
(10) 
that is, the number of ways of arranging elements in trees rooted on a permutation of a subset of (recall that is the number of all possible permutations of elements). Now to obtain the number of ways of arranging elements in trees rooted on even cycles, each of which is a hung majority we adapt (10) as follows. First we establish the number of delegation graphs (for a given issue) which contain only even cycles, that is:
(11) 
If each addendum of the above expression is multiplied by , that is the number of possible opinions on of agents, one obtains the number of possible proxy profiles with default that determine a delegation graph with only even cycles, with all the possible assignments of opinions for the agents in the permutation on which the trees of the graph are rooted:
(12) 
We can then adapt (12) by restricting the subprofiles of opinions of the agents to hung majorities (i.e., ). We thus obtain the following value:
(13) 
Under the impartial culture assumption, the probability of a proxy profile with default opinions to induce only even cycles with hung majorities is therefore (13) divided by . This quantity approaches as tends to infinity. ∎
Proof of Theorem 3.
Assume that for all , for all such that is a cycle in , for all : . Consider an arbitrary . Let be the distance from to the closest agent in a cycle of , and let denote . We show that for any , is an opinion which will not change at any later stage (stable).

If : is its only infuencer, therefore is stable by assumption.

If : Assume (IH) that for all agents such that , is stable. This implies that all influencers of are stable. There are two cases:

If is not consistent, then it will never be, and therefore is already stable.

If is consistent, then for each , , and is therefore (by IH) stable.

It follows that after steps, with , each agent’s opinion is stable, and the BDP has therefore stabilized. ∎
Proof of Theorem 4.
We proceed by contraposition. Let , be a cycle in , , and . Let be the length of the cycle and be the distance from to . Then will enter a loop of size : for all , . Therefore, the BDP does not stabilize. Assume be such that is a cycle in , and for all , . Then, for all , and all
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