1 Introduction
Liquid democracy [liquid_feedback] is a form of democratic decisionmaking considered to stand between direct and representative democracy. It has been used, advocated and popularized by local and even national parties (e.g., Demoex^{2}^{2}2demoex.se/en/ in Sweden, and Piratenpartei^{3}^{3}3www.piratenpartei.de in Germany) to coordinate the behavior of party representatives in assemblies, as well as campaigns (e.g., Make Your Laws^{4}^{4}4www.makeyourlaws.org in the US). At its heart is voting via a delegable proxy, also called sometimes 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, or adaptations thereof), but their votes now carry a weight consisting of the number of all agents that, directly or indirectly, entrusted them with their vote.
Scientific context and contribution
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]. Delegable proxy has not, to the best of our knowledge, been object of study so far, with the notable exception of [Boldi_2011] which focuses specifically on algorithmic aspects of a variant of liquid democracy (which the authors refer to as viscous democracy) with applications to recommender systems.
The objective of the paper is to provide a first analysis, via formal methods, of the liquid democracy voting system based on delegable proxy. This, we hope, should point to a number of future lines of research and stimulate further investigations into this and related systems.
Outline
The paper starts in Section 2 by introducing some preliminaries on the theory of binary aggregation, which is the framework of reference for this study. It is then structured in two parts. This preliminary section presents also novel results on binary aggregation with abstentions. The first part (Section 3) studies voting in liquid democracy from the point of view of the delegation of voting power: we study delegable proxy aggregators using the machinery of binary and judgment aggregation. This allows us to shed novel light on some issues involved in the liquid democracy system, in particular: the issue of circular delegation, and the issue of individual irrationality when voting on logically interdependent issues. The second part (Sections 4 and 5) studies voting in liquid democracy as a very specific type of opinion diffusion on networks, whereby delegation is rather interpreted as the willingness to copy the vote of a trustee. We show that this perspective provides some interesting insights on how to address the above mentioned issues of circular delegations and individual irrationality. Section 6 concludes the paper and outlines some ongoing lines of research.
2 Binary Aggregation with Abstention
The formalism of choice for this paper is binary aggregation [grandi13lifting] with abstention.^{5}^{5}5The standard framework of binary aggregation without abstention is sketched in the appendix for ease of reference. This preliminary 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 s.t. ;

is a nonempty finite set of issues (), each represented by a propositional atom;

is an (integrity) constraint, where is the propositional language constructed by closing under a functionally complete set of Boolean connectives (e.g., ).
An opinion function is an assignment acceptance/rejection values (or, truth values) to the set of issues . 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 . Allowing abstention in the framework of binary aggregation 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.
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 . As the latter are the opinions we are interested in, we will often refer to them simply as individual opinions.
An opinion profile records the opinion, on the given set of issues, of every individual in . Given a profile the projection is denoted (i.e., the opinion of agent in profile ). Let us introduce some more notation. We also denote by the set of agents accepting issue in profile , by and by the set of nonabstaining agents. We write to denote that the two profiles and are identical, except for possibly the opinion of voter .
2.2 Aggregators
Given a BA structure , an aggregator (for ) is a function , mapping every profile of individual opinions to one collective (possibly incomplete) opinion.^{6}^{6}6It is therefore worth stressing that, in this paper, we study aggregators that are resolute (that is, output exactly one value), even though they allow for collective abstention. denotes the outcome of the aggregation on issue . The benchmark aggregator is the issuebyissue strict majority rule (), which 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. In general, quota rules in binary aggregation with abstention are of the form: accept when the proportion of nonabstaining individuals accepting is above the acceptancequota, reject when the proportion of nonabstaining individuals is above the rejectionquota, and abstain otherwise:^{7}^{7}7There are several ways to think of quota rules in the presence of 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 an aggregation structure. A quota rule (for ) is defined as follows, for any issue , and any opinion profile :^{8}^{8}8The definition uses the ceiling function denoting the smallest integer larger than .
(2) 
where 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.^{9}^{9}9Those 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 (3) the above defines an aggregator of type as desired.^{10}^{10}10What 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 a given issue Notice also that if the rule is symmetric, then (3) forces .
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 . 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 natural uniform but asymmetric variant of unanimity can be obtained by setting and .
Let us finally note an important difference between quota rules in binary aggregation with abstentions vs. without abstentions. In a framework without abstentions quota rules are normally defined by a unique acceptance quota , the rejection quota being uniquely determined as . As a consequence, the majority rule, when
is odd, is the only unbiased quota rule in the standard framework. This is no longer the case when abstentions are considered. A novel characterization of the majority rule will be given in Section
2.5.2.3 Agenda conditions
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 conditions. We provide just a simple illustrative example.
Example 3.
Let and let . is not simple. The set is inconsistent with , but none of its subsets is.
2.4 Properties of aggregators
We start by recalling some wellknown properties of aggregators from the judgment aggregation literature, adapted to the setting with abstention:
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.
 dictatorial

iff there exists (the dictator) s.t. for any profile , and all , iff . I.e., there exists an agent whose definite opinion determines the group’s definite opinion on . If is dictatorial, with the same dictator on all issues , then it is called dictatorial.
 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.
 systematic

iff it is neutral and independent. I.e., the collective opinion on issue depends only on the individual opinions on this issue.
 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 individual opinions on an issue (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.
Example 4.
It is wellknown that majority is not rational in general. The standard example is provided by the socalled discursive dilemma, represented by the BA structure . The profile consisting of , , , returns an inconsistent majority opinion (cf. [Grossi_2014]).
Finally, let us defined also 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 :
(4) 
2.5 Characterizing quota rules
As a typical example, consider the aggregator : it is unanimous, anonymous, monotonic, systematic, responsive and unbiased, but, as mentioned above, it is not rational in general. However, it can be shown (cf. [Grossi_2014, 3.1.1]) that aggregation by majority is collectively rational under specific assumptions on the constraint:
Fact 1.
Let be a BA structure with a simple agenda. Then is rational.
Proof.
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. ∎
May’s theorem [May_1952] famously shows that for preference aggregation, the majority rule is in fact the only aggregator satisfying a specific bundle of desirable properties. A corresponding characterization of the majority rule is given in judgment aggregation without abstention: 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 abstention (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 .
Proof.
The claim is proven by the following series of equivalent statements. (a) A uniform and symmetric quota rule has quota such that . (b) A uniform and symmetric quota rule has quota such that for any profile and issue . (c) For any and , if and only if , that is, an even number of voters vote and the group is split in half. (d) , 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.
Proof.
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.
By the above theorem and Fact 1, it follows that, on simple agendas, majority is the only rational aggregator which is also responsive, anonymous, systematic and monotonic.
2.6 Impossibility in Binary Aggregation with Abstentions
The following is a wellknow 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.
We will use this result to illustrate how impossibility results from binary aggregation with abstentions apply to delegable proxy voting on binary issues.
3 Liquid Democracy as Binary Aggregation
In this section we provide an analysis of liquid democracy by embedding it in the theory of binary aggregation with abstentions presented in the previous section. To the best of our knowledge, this is the first attempt at providing an analysis of delegable proxy voting using socialchoice theoretic tools, with the possible exception of [greenarmytage_delegable].
In what follows we will often refer to delegable proxy voting/aggregation simply as proxy voting/aggregation.
3.1 Binary Aggregation via Delegable Proxy
In binary aggregation with proxy, agents either express an acceptance/rejection opinion or delegate such opinion to a different agent.
3.1.1 Proxy Opinions and Profiles
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 opinion to be individually rational, in a precise sense (Section 3.2.2). For simplicity we are assuming that abstention is not a feasible opinion in proxy voting, but that is an assumption that 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 consistent proxy opinions, being the set of all proxy profiles.
3.1.2 Delegation Graphs
Each profile of proxy opinions (proxy profile in short) 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 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 ):^{11}^{11}11 We recall that the reflexive transitive closure of a binary relation is the smallest reflexive and transitive relation that contains . . 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 and we will come back to this issue later.^{12}^{12}12See also footnote 16 below.
For all , we also define the function such that . 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 , that is, 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 on is a profile where no voter expresses an opinion on , 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 .
3.1.3 Proxy Aggregators
The most natural form of voting via delegable proxy is a proxy version of the majority rule we discussed in Section 2:^{13}^{13}13On the importance of majority decisions in the current implementation of liquid democracy by liquid feedback cf. [liquid_feedback, p.106].
(6) 
That is, 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. It should be clear 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) .
In general, it should be clear that for any quota rule a proxy variant of can be defined via an obvious adaptation of (6).
To fix intuitions further about proxy voting it is worth discussing another example of aggregator, proxy dictatorship. It is defined as follows, for a given (the dictator) any proxy profile and issue :
(7) 
That is, in a proxy dictatorship, the collective opinion is the opinion of the guru of the dictator, when it exists, and it is undefined otherwise.
3.2 Two Issues of Delegable Proxy
3.2.1 Cycles and Abstentions
It should be clear from the definition of proxy aggregators like , that such 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.
Fact 2.
Let be a BA structure where (i.e., issues are independent). If each proxy profile is equally probable (impartial culture assumption), then the probability that, for each issue , the delegation graph has no gurus tends to as tends to infinity.
Proof.
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) is . The claimed value is then established through this series of equations:
This completes the proof. ∎
Now contrast the above simple fact 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 tends to as tends to infinity.
Fact 2
should obviously not be taken as a realistic estimate of the effect of cycles on collective abstention, as the impartial culture assumption is a highly idealized assumption. Election data should ideally be used to assess whether delegation cycles ever lead large parts of the electorate to ’lose their vote’, possibly together with refinements of the above argument that take into consideration realistic distributions on proxy profiles, and therefore realistic delegation structures. Nonetheless, Fact
2 does flag a potential problem of cyclical delegations as sources of abstention which has, to the best of our knowledge, never been discussed. The mainstream position on cyclical delegations [liquid_feedback, Section 2.4.1] is:^{14}^{14}14Cf. also [Behrens15].“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.”
We will discuss later (Section 4) a possible approach to mitigate this issue by suggesting a different interpretation of liquid democracy in terms of influence rather than delegation.
3.2.2 Individual & 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 this section 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
(8) 
is satisfiable (consistency), and if whenever (8) entails , then belongs to it (closedness).^{15}^{15}15Cf. the definition of individual opinions in Section 2. 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 (w.r.t. the available issues).
The constraint in (8) captures, one might say, an idealized way of how delegation works: voters are assumed to be able to check or monitor how their gurus are voting, and always modify their delegations if an inconsistency arises. The constraint remains, however, rather counterintuitive under a delegative interpretation of proxy voting. Aggregation via delegable proxy has at least the potential to represent individual opinions as irrational (inconsistent and/or not logically closed).
Like in the case of delegation cycles we will claim that the interpretation of liquid democracy in terms of influence to be developed in Section 4, rather than in terms of delegation, makes individual rationality at least as defensible as in the classical case.
3.3 Embedding in Binary Aggregation with Abstentions
3.3.1 One man—One vote
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:
“[…] 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. 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).^{16}^{16}16It should be clear that not every proxy aggregator satisfies this 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 the following diagram:
which gives us a handle to study a large class of proxy voting rules
Example 5.
It follows that for every proxy aggregator the axiomatic machinery developed for standard aggregators can be directly tapped into. Characterization results then extend effortlessly to proxy voting, again providing a strong rationale for the use of majority in proxy aggregation:
Fact 3 (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.
Proof.
This follows from the definition of and Theorem 1. ∎
It follows that on simple agendas and assuming the individual rationality of proxy profiles, proxy majority is the only rational aggregator which is anonymous, independent, monotonic, responsive, neutral and minimizes undecisiveness.
3.3.2 Impossibility
Similarly, there are many ways in which pursue the opposite embedding, from standard aggregation into proxy voting. 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, where . The function is defined as follows, for any and : if , , otherwise.^{17}^{17}17Notice 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. A dummy voter can be introduced for that purpose. A translation of this type allows to think of standard aggregators as the concatenation , for some proxy aggregator :
Fact 4.
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.
Proof.
It follows directly from the definition of and Theorem 2. ∎
3.4 Section Summary
The section has provided a very simple model of delegable proxy voting within the framework of binary aggregation. This has allowed us to put liquid democracy in perspective with an established body of results in the social choice theory tradition, and highlight two of its problematic aspects, which have so far gone unnoticed: the effect of cycles on collective indecisiveness, and the issue of preservation of individual rationality under delegable proxies.
An independent, purposebuilt axiomatic analysis for liquid democracy focused on its more characteristic features (like the one man—one vote property) is a natural line of research, which we do not pursue here.
4 Liquid Democracy as Binary Opinion Diffusion
Proxy voting can also be studied from a different perspective. Imagine a group where, for each issue , each agent copies the opinion of a unique personal “guru”. 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. The collective opinion of a group of agents who either express a opinion or delegate 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 opinion but gets there by copying the opinion of some unique “guru” (possibly themselves). In this perspective, a proxy voting aggregation can be assimilated to a (converging) process of opinion formation.
The above interpretation of liquid democracy is explicitly put forth in [liquid_feedback]:
“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].
The current section develops an analysis of this interpretation, and highlights some of its advantages over the delegationbased one studied earlier.
4.1 Binary aggregation and binary influence
The section develops a very simple model of binary influence based on the standard framework of binary aggregation (see Appendix B for a concise presentation). For 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.
4.1.1 DeGroot Processes and Opinion Diffusion
In [Degroot_1974], DeGroot proposes a simple model of stepbystep opinion change under social influence. The model combines two types of matrices. Assuming a group of agents, a first matrix represents the weighted influence network (who influences whom and how much), and a second matrix represents the probability assigned by each agent to each of the different alternatives. Both the agents’ opinion and the influence weights are taken within and are (row) stochastic (each row sums up to ). Given an opinion and an influence matrix, the opinion of each agent in the next time step is obtained through linear averaging.
Here we focus on a specific class of opinion diffusion processes in which opinions are binary, and agents are influenced by exactly one influencer, possibly themselves, of which they copy the opinion. The model captures a class of processes which lies at the interface of two classes of diffusion models that have remained so far unrelated: the stochastic opinion diffusion model known as DeGroot’s [Degroot_1974], and the more recent propositional opinion diffusion model due to [Grandi:2015:POD:2772879.2773278]. The diffusion processes underpinning liquid democracy—which we call here Boolean DeGroot processes (BDPs)—are the special case of the DeGroot stochastic processes and, at the same time, the special case of propositional opinion diffusion processes where each agent has access to the opinion of exactly one neighbor (cf. Figure 1).
4.1.2 Boolean DeGroot processes
Here we focus on the Boolean special case of a DeGroot process showing its relevance for the analysis of liquid democracy. Opinions are defined over a BA structure, and hence are taken to be binary. Similarly, we take influence to be modeled by the binary case of an influence matrix. Influence is of an “allornothing” type and each agent is therefore taken to be influenced by exactly one agent, possibly itself. The graph induced by such a binary influence matrix (called influence graph) is therefore a structure where is a binary relation where is taken here to denote that “ is influenced by ”. Such relation is serial () and functional ( if and then ). So each agent has exactly one successor (the influencer), possibly itself, which we denote . It should be clear that influence graphs are the same sort of structures we studied earlier in Section 3 under the label ’delegation graph’.
An influence profile records how each agent is influenced by each other agent, with respect to each issue . Given a profile the i projection denotes the influence graph for issue , also written .
So let us define the type of opinion dynamics driving BDPs:
Definition 4 (Bdp).
Now fix an opinion profile and an influence profile . Consider the stream of opinion profiles recursively defined as follows:

Base:

Step: for all , , .
where . We call processes defined by the above dynamics Boolean DeGroot processes (BDPs).
It should be clear that the above dynamics is the extreme case of linear averaging applied on binary opinions and binary influence.
As noted above, BDPs are also the special case of processes that have recently been proposed in the multiagent systems literature as propositional opinion diffusion processes [Grandi:2015:POD:2772879.2773278], i.e., cases where 1) the aggregation rule is the unanimity rule (an agent adopts an opinion if and only if all her influencers agree on it), and 2) each agent has exactly one influencer. We will come back to the link with propositional opinion diffusion in some more detail later in Section 4.3.
4.2 Convergence of BDPs
When do the opinions of a group of individuals influencing each other stabilize? Conditions have been given, in the literature, for the general paradigms of which BDPs are limit cases. This section introduces the necessary graphtheoretic notions and briefly recalls those results before giving a characterization of convergence for BDPs.
4.2.1 Preliminaries
We start with some terminology. We say that the stream of opinion profiles converges if there exists such that for all , if , then .
We will also say that a stream of opinion profiles converges for issue if there exists such that, for all , if , then . Given a stream of opinion profiles starting at we say that agent stabilizes in that stream for issue if there exists such that for any . So a BDP on influence graph starting with the opinion profile is said to converge if the stream generated according to Definition 4 where converges. Similarly, A BDP is said to converge for issue if its stream converges for , and an agent in the BDP is said to stabilize for if it stabilizes for in the stream generated by the BDP.
Notice first of all that influence graphs have a special shape:^{18}^{18}18Please consult Appendix B for the relevant terminology from graph theory.
Fact 5.
Let be an influence graph and be a connected component of . Then contains exactly one cycle, and the set of nodes in the cycle is closed.
Proof.
Assume that does not contain any cycle. Since is finite and since no path can repeat any node, any path in is finite too. Let be the last element of (one of) the longest path(s) in . Then does not have any successor, which contradicts seriality. So contains at least one cycle. Let be the set of nodes of a cycle in . Assume that is not closed: for some and , . Since is a cycle, there is also some , such that , which contradicts functionality. Therefore, the nodes of any cycle in forms a closed set. Now assume that contains more than one cycle. Since the nodes of each cycle forms a closed set, there is no path connecting any node inside a cycle to any node in any other cycle, which contradicts connectedness. So contains a unique cycle, whose nodes form a closed set. ∎
Intuitively, influence graphs of BDPs then look like sets of confluent chains aiming together towards common cycles.
4.2.2 Context: convergence in DeGroot processes
For the general case of DeGroot processes, an influence structure guarantees that any distribution of opinions will converge if and only if “every set of nodes that is strongly connected and closed is aperiodic” [jackson08social, p.233]. In the propositional opinion diffusion setting, sufficient conditions for stabilization have been given by [Grandi:2015:POD:2772879.2773278, Th. 2]: on influence structures containing cycles of size at most one (i.e, only selfloops), for agents using an aggregation function satisfying (ballot)monotonicity and unanimity^{19}^{19}19Notice that the rule underpinning BDP, that is the ‘gurucopying’ rule on serial and functional graphs, trivially satisfies those constraints., opinions will always converge in at most at most steps, where is the diameter of the graph.^{20}^{20}20A second sufficient condition for convergence is given by [Grandi:2015:POD:2772879.2773278]: when agents use the unanimity aggregation rule, on irreflexive graphs with only vertexdisjoint cycles, such that for each cycle there exists an agent who has at least two influencers, opinions converge after at most steps. Note that no BDP satisfies this second condition. The results below show how BDPs are an interesting limit case of both DeGroot and propositional opinion diffusion processes.
4.2.3 Two results
It must be intuitively clear that nonconvergence in a BDP is linked to the existence of cycles in the influence graphs. However, from the above observation (Fact 5), we know that nodes in a cycle cannot have any influencers outside this cycle, and hence that cycles (including selfloops) can only occur at the “tail” of the influence graph. Hence, if the opinions in the (unique) cycle do not converge, which can only happen in a cycle of length , the opinions of the whole population in the same connected component will not converge. The above implies that for any influence graphs with a cycle of length , there exists a distribution of opinions which loops. This brings us back to convergence result for general (not necessarily Boolean) DeGroot processes. Indeed, for functional and serial influence graphs, a closed connected component is aperiodic if and only if its cycle is of length .
Fact 6.
Let be an influence profile. Then the following are equivalent:

The BDP converges for any opinion profile on .

For all , contains no cycle of length .

For all , all closed connected components of are aperiodic.
Proof.
Let and assume that contains no cycle of length and has diameter . Let be a connected component of . By Fact 5, contains a unique cycle, which, by assumption, is of length . Hence, is aperiodic. Let be the node in the cycle. The opinion of will spread to all nodes in after at most steps. Therefore, all BDPs on will converge after at most steps, where is the maximum within the set of diameters of for all . We proceed by contraposition. Assume that for some , a connected component of contains a cycle of length . By 5, this cycle is unique, and therefore the greatest common divisor of the cycles lengths of is , so is not aperiodic. Let be the set of nodes in the cycle. Let be such that for some with distance from to , . Then will not converge, but enter a loop of size : for all , . Hence, does not converge. Trivial. ∎
It is worth noticing that one direction (namely from to ) of the above result is actually a corollary of both the convergence result for DeGroot processes stated at the beginning of this section (cf. [jackson08social]), and of a known convergence result for propositional opinion diffusion [Grandi:2015:POD:2772879.2773278, Th. 2], also stated earlier.
The above gives a characterization of the class of influence profiles on which all opinion streams converge. But we can aim at a more general result, characterizing the class of pairs of opinion and influence profiles which lead to convergence:
Theorem 3.
Let be an influence profile and be an opinion profile. Then the following statements are equivalent:

The BDP converges for on .

For all , there is no set of agents such that: is a cycle in and there are two agents such that .
Proof.
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 , . Assume be such that is a cycle in , and for all , . Then, for all , and all , and for all with distance from to , for all , such that , . ∎
This trivially implies that the class of opinion profiles which guarantees convergence for any influence profile, is the one where everybody agrees on everything already. Note that the only stable distributions of opinions are the ones where, in each connected component in , all members have the same opinion, i.e, on BDPs, converging and reaching a consensus (within each connected component) are equivalent, unlike in the stochastic case. Moreover, for an influence profile where influence graphs have at most diameter and the smallest cycle in components with diameter is of length , it is easy to see that if a consensus is reached, it will be reached in at most steps, which is at most .
4.2.4 Liquid Democracy as a BDP
We have seen (Section 3) that each proxy profile induces what we called a delegation graph for each issue . Delegation graphs are the same sort of structures we referred to in the current section as influence graphs. So each proxy profile can be associated to a BDP by simply assigning random or opinions to each voter delegating her vote in . It is then easy to show that for each connected component of , if has a guru with opinion , then that component stabilizes in the BDP on opinion for each assignment of opinions to the delegating agents in . Vice versa, if stabilizes on value in the BDP for each assignment of opinions to the delegating agents in , then has a guru whose opinion is . This establishes a direct correspondence between voting with delegable proxy and Boolean deGroot processes. However, BDPs offer an interesting and novel angle on the issue of cyclical delegations, to which we turn now.
4.2.5 Cycles
As discussed earlier (Section 3.2.1), cycles are a much discussed issue in liquid democracy. Its proponents tend to dismiss delegation cycles as a nonissue: since the agents forming a cycle delegate their votes, none of them is casting a ballot and the cycles get resolved essentially by not counting the opinions of the agents involved in the cycle [liquid_feedback]. We stressed this solution as problematic in the ‘votedelegation’ interpretation of liquid democracy as it has the potential to discard large numbers of opinions. The elimination of cycles not only hides to aggregation the opinions of the agents involved in cycles, but also the opinions of agents that may be linked to any of those agents by a delegation path. In other words information about entire connected components in the delegation graph may be lost.
We argue that the ‘votecopying’ interpretation of the system—formalized through BDPs—offers novel insights into possible approaches to cycles in delegable proxy. Theorems 3 and 6 offer an alternative solution by showing that not all cycles are necessarily bad news for convergence: cycles in which all agents agree still support convergence of opinions, and therefore a feasible aggregation of opinions by proxy. This suggests that alternative proxy voting mechanisms could be designed based on opinion convergence behavior rather than on weighted voting.
4.3 Excursus: unanimity and 2colorability
In the above, we have worked at the intersection of two models of opinion diffusion, the DeGroot model, and the propositional opinion diffusion model. However, there is more to say about how the two frameworks relate.
Let us take a brief detour towards a generalisation of BDPs corresponding to the case of propositional opinion diffusion with the unanimity rule, where agents can have several influencers and change their opinions only if all their influencers disagree with them. This means that we relax the functionality constraint on influence graphs. We will show how the two frameworks meet again: some nonstabilizing opinion cases under the unanimity rule correspond to a special class among the ‘semiBoolean’ cases of DeGroot processes where opinions are still binary but influence does not need to be.
We define the dynamics of opinions under the unanimity rule in the obvious way:
Definition 5 (Up).
Fix an opinion profile and a (serial but nonnecessarily functional) influence profile . Consider the stream of opinion profiles recursively defined as follows:

Base:

Step: for all and all :
(9)
where . We call processes defined by the above dynamics Unanimity Processes (UPs).
We give a sufficient condition for nonconvergence of UPs:
Lemma 2.
Let be a (serial and nonnecessarily functional) influence profile and be an opinion profile, such that, for some , for all , where is a connected component of : if , then . Then does not converge in UP.
Proof.
Let be a (serial and nonnecessarily functional) influence profile, and be an opinion profile, such that, for some , for all with a connected component of : if , then . Then, by definition of UPs, for all , , and by repeating the same argument, for all , . ∎
Intuitively, the above condition for nonconvergence corresponds to a situation of global maximal disagreement: all agents (of a connected component) disagree with all their influencers. Recall that a graph is properly colored if each node is assigned exactly one among colors and no node has a successor of the same color, and consider the two possible opinions on issue
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